Calculus II Sections 9.5, 10.3: Finding and Using Taylor Series, - - PowerPoint PPT Presentation

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Calculus II

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 April 21, 2020 (Section 9.5 WebWork Mini-Set due 11:59 PM, April 22nd)

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

1 Apply the Ratio Test on ck(x − a)k by calculating

lim

k→+∞

• ck+1(x − a)k+1

ck(x − a)k

• for x = a.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

1 Apply the Ratio Test on ck(x − a)k by calculating

lim

k→+∞

• ck+1(x − a)k+1

ck(x − a)k

• for x = a.

2 The power series converges absolutely when the above limit is

less than 1 and diverges when the above limit is greater than 1.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

1 Apply the Ratio Test on ck(x − a)k by calculating

lim

k→+∞

• ck+1(x − a)k+1

ck(x − a)k

• for x = a.

2 The power series converges absolutely when the above limit is

less than 1 and diverges when the above limit is greater than 1.

3 The ratio test is inconclusive for any x values where the above

limit is exactly 1. In this case, you will need to apply a different convergence test for such x values individually.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Process for Finding Radii and Intervals of Convergence

The general procedure to find the radius and interval of convergence of a power series is as follows:

1 Apply the Ratio Test on ck(x − a)k by calculating

lim

k→+∞

• ck+1(x − a)k+1

ck(x − a)k

• for x = a.

2 The power series converges absolutely when the above limit is

less than 1 and diverges when the above limit is greater than 1.

3 The ratio test is inconclusive for any x values where the above

limit is exactly 1. In this case, you will need to apply a different convergence test for such x values individually.

Reminder: the interval of convergence is centered about x = a and contains the interval (a − R, a + R) at minimum.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

Find the radius and interval of convergence of

∞

• n=0

n(x + 2)n 3n+1 .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

Find the radius and interval of convergence of

∞

• n=0

n(x + 2)n 3n+1 . Apply the Ratio Test. lim

n→+∞

• (n + 1)(x + 2)n+1

3n+2 · 3n+1 n(x + 2)n

• =

lim

n→+∞

• (n + 1)(x + 2)

3n

• = |x + 2|

3 · lim

n→+∞

n + 1 n = |x + 2| 3

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

So the power series converges absolutely if |x + 2| 3 < 1 ⇒ |x + 2| < 3 ⇒ −3 < x + 2 < 3 ⇒ −5 < x < 1. (Since the power series is centered about x = −2, it follows that the radius of convergence R is 3.)

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

So the power series converges absolutely if |x + 2| 3 < 1 ⇒ |x + 2| < 3 ⇒ −3 < x + 2 < 3 ⇒ −5 < x < 1. (Since the power series is centered about x = −2, it follows that the radius of convergence R is 3.) The power series diverges if |x + 2| 3 > 1, i.e. x < −5 or x > 1.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

So the power series converges absolutely if |x + 2| 3 < 1 ⇒ |x + 2| < 3 ⇒ −3 < x + 2 < 3 ⇒ −5 < x < 1. (Since the power series is centered about x = −2, it follows that the radius of convergence R is 3.) The power series diverges if |x + 2| 3 > 1, i.e. x < −5 or x > 1. For |x + 2| 3 = 1 (that is, when x = −5 or x = 1), the Ratio Test fails. So we need to investigate these cases separately.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

x = −5 : The power series becomes

∞

• n=0

n(−5 + 2)n 3n+1 =

∞

• n=0

n · (−3)n 3n+1 =

∞

• n=0

n · (−1)n 3 , which diverges since the terms do not converge to 0.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

x = −5 : The power series becomes

∞

• n=0

n(−5 + 2)n 3n+1 =

∞

• n=0

n · (−3)n 3n+1 =

∞

• n=0

n · (−1)n 3 , which diverges since the terms do not converge to 0. x = 1 : The power series becomes

∞

• n=0

n(1 + 2)n 3n+1 =

∞

• n=0

n · 3n 3n+1 =

∞

• n=0

n 3, which diverges since the terms do not converge to 0.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• n=0

n(x+2)n 3n+1

(cont’d)

x = −5 : The power series becomes

∞

• n=0

n(−5 + 2)n 3n+1 =

∞

• n=0

n · (−3)n 3n+1 =

∞

• n=0

n · (−1)n 3 , which diverges since the terms do not converge to 0. x = 1 : The power series becomes

∞

• n=0

n(1 + 2)n 3n+1 =

∞

• n=0

n · 3n 3n+1 =

∞

• n=0

n 3, which diverges since the terms do not converge to 0. So the radius of convergence is R = 3 and the interval of convergence is (−5, 1) (or −5 < x < 1).

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

What If the Power Series Is Geometric?

If a power series is a geometric series, i.e. of the form

∞

• k=0

ark, then the general procedure used to compute the radius and interval of convergence is unnecessary.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

What If the Power Series Is Geometric?

If a power series is a geometric series, i.e. of the form

∞

• k=0

ark, then the general procedure used to compute the radius and interval of convergence is unnecessary. We can find the interval of convergence simply by recalling that

∞

• k=0

ark converges (absolutely!) if and only if |r| < 1.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• k=0

(−1)k(2

5)k(x + 3)k

Find the interval of convergence of

∞

• k=0

(−1)k 2 5 k (x + 3)k.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• k=0

(−1)k(2

5)k(x + 3)k

Find the interval of convergence of

∞

• k=0

(−1)k 2 5 k (x + 3)k. This is a geometric series with a = 1 and r = −2 5(x + 3). (Note that r is a function of x.) It converges if and only if −1 < −2 5(x + 3) < 1 ⇒ −5 2 < x + 3 < 5 2 ⇒ −11 2 < x < −1 2

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example:

∞

• k=0

(−1)k(2

5)k(x + 3)k

Find the interval of convergence of

∞

• k=0

(−1)k 2 5 k (x + 3)k. This is a geometric series with a = 1 and r = −2 5(x + 3). (Note that r is a function of x.) It converges if and only if −1 < −2 5(x + 3) < 1 ⇒ −5 2 < x + 3 < 5 2 ⇒ −11 2 < x < −1 2 So the radius of convergence is R = 5 2 and the interval of convergence is

• −11

2 , −1 2

• .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

If a function is expressed as a power series, then its domain is just the interval of convergence of the series.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

If a function is expressed as a power series, then its domain is just the interval of convergence of the series. So for instance, if f (x) =

∞

• k=0

(−1)k 2 5 k (x + 3)k, then the domain of f is

• −11

2 , −1 2

• .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Harder Example: A Bessel Function

Find the domain of the following Bessel function. J0(x) =

∞

• n=0

(−1)n · x2n 22n(n!)2

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Harder Example: A Bessel Function

Find the domain of the following Bessel function. J0(x) =

∞

• n=0

(−1)n · x2n 22n(n!)2 Apply the Ratio Test. lim

n→+∞

• an+1

an

• =

lim

n→+∞

• (−1)n+1 · x2n+2

22n+2((n + 1)!)2 · 22n(n!)2 (−1)n · x2n

• =

lim

n→+∞

• −x2

22(n + 1)2

• = x2

4 · lim

n→+∞

1 (n + 1)2 = 0

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Harder Example: A Bessel Function (cont’d)

Since lim

n→+∞

• an+1

an

• is identically 0 < 1, regardless of what x

is, the power series converges for all x.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Harder Example: A Bessel Function (cont’d)

Since lim

n→+∞

• an+1

an

• is identically 0 < 1, regardless of what x

is, the power series converges for all x. In such situations, the radius of convergence is R = +∞.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Harder Example: A Bessel Function (cont’d)

Since lim

n→+∞

• an+1

an

• is identically 0 < 1, regardless of what x

is, the power series converges for all x. In such situations, the radius of convergence is R = +∞. As a result, the interval of convergence/domain of the Bessel function is (−∞, +∞).

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Last time, we raised the question of whether a power series is a Taylor series.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Last time, we raised the question of whether a power series is a Taylor series. In other words, given a power series

∞

• n=0

cn(x − a)n, is there a function f (x) whose Taylor series about x = a is the power series?

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Last time, we raised the question of whether a power series is a Taylor series. In other words, given a power series

∞

• n=0

cn(x − a)n, is there a function f (x) whose Taylor series about x = a is the power series? The answer is yes! Here’s the first step to seeing why.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

1 f is differentiable on (x0 − R, x0 + R) and

f ′(x) =

∞

• k=1

kck(x − x0)k−1.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

1 f is differentiable on (x0 − R, x0 + R) and

f ′(x) =

∞

• k=1

kck(x − x0)k−1.

2 f ′(x) also has a radius of convergence of R.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof Idea

Naively, we would like for f ′(x) = d dx ∞

• k=0

ck(x − x0)k

• =

∞

• k=0

d dx

• ck(x − x0)k

.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof Idea

Naively, we would like for f ′(x) = d dx ∞

• k=0

ck(x − x0)k

• =

∞

• k=0

d dx

• ck(x − x0)k

. By power rule, this would yield f ′(x) =

∞

• k=0

kck(x − x0)k−1 =

∞

• k=1

kck(x − x0)k−1, which can be shown has the same radius of convergence R by the Ratio Test or Root Test.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof Idea

Naively, we would like for f ′(x) = d dx ∞

• k=0

ck(x − x0)k

• =

∞

• k=0

d dx

• ck(x − x0)k

. By power rule, this would yield f ′(x) =

∞

• k=0

kck(x − x0)k−1 =

∞

• k=1

kck(x − x0)k−1, which can be shown has the same radius of convergence R by the Ratio Test or Root Test. One can then appeal to the definition of the derivative and the absolute convergence of power series inside their intervals

• f convergence to see that the above formula does hold.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation

One differentiates a power series “term-by-term”.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation

One differentiates a power series “term-by-term”. So if you have that f (x) =

∞

• k=0

ck(x−x0)k = c0+c1(x−x0)+c2(x−x0)2+c3(x−x0)3+· · · , then you can just keep using power rule to get that f ′(x) =

∞

• k=1

kck(x−x0)k−1 = 0+c1+2c2(x−x0)+3c3(x−x0)2+· · · .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation

One differentiates a power series “term-by-term”. So if you have that f (x) =

∞

• k=0

ck(x−x0)k = c0+c1(x−x0)+c2(x−x0)2+c3(x−x0)3+· · · , then you can just keep using power rule to get that f ′(x) =

∞

• k=1

kck(x−x0)k−1 = 0+c1+2c2(x−x0)+3c3(x−x0)2+· · · . We can now answer our earlier question!

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation in Action

For instance, given that we know the Maclaurin series for sin x is

∞

• k=0

(−1)k (2k + 1)!x2k+1 = x − x3 3! + x5 5! − x7 7! + · · · we can find the Maclaurin series for cos x via term-by-term differentiation.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation in Action

For instance, given that we know the Maclaurin series for sin x is

∞

• k=0

(−1)k (2k + 1)!x2k+1 = x − x3 3! + x5 5! − x7 7! + · · · we can find the Maclaurin series for cos x via term-by-term differentiation. d dx ∞

• k=0

(−1)k (2k + 1)!x2k+1

• =

∞

• k=0

(−1)k (2k + 1)!(2k + 1)x2k =

∞

• k=0

(−1)k (2k)! x2k = 1 − x2 2! + x4 4! − x6 6! + · · · ,

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Differentiation in Action

For instance, given that we know the Maclaurin series for sin x is

∞

• k=0

(−1)k (2k + 1)!x2k+1 = x − x3 3! + x5 5! − x7 7! + · · · we can find the Maclaurin series for cos x via term-by-term differentiation. d dx ∞

• k=0

(−1)k (2k + 1)!x2k+1

• =

∞

• k=0

(−1)k (2k + 1)!(2k + 1)x2k =

∞

• k=0

(−1)k (2k)! x2k = 1 − x2 2! + x4 4! − x6 6! + · · · , which we do already recognize as the Maclaurin series for cos x.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Corollary Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then f (x) is smooth over the interval (x0 − R, x0 + R). Furthermore, f (x) is equal to its own Taylor series about x = x0.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof of the Corollary

Since the derivative of a power series is itself a power series with the exact same radius of convergence, it is clear that all power series are infinitely-times differentiable.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof of the Corollary

Since the derivative of a power series is itself a power series with the exact same radius of convergence, it is clear that all power series are infinitely-times differentiable. Repeated differentation of f (x) =

∞

• k=0

ck(x − x0)k yields f (n)(x) =

∞

• k=n

k(k − 1)(k − 2) · · · (k − n + 1)ck(x − x0)k−n.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Proof of the Corollary

Since the derivative of a power series is itself a power series with the exact same radius of convergence, it is clear that all power series are infinitely-times differentiable. Repeated differentation of f (x) =

∞

• k=0

ck(x − x0)k yields f (n)(x) =

∞

• k=n

k(k − 1)(k − 2) · · · (k − n + 1)ck(x − x0)k−n. In particular, f (n)(x0) = n!cn from which it follows that cn = f (n)(x0) n! , as desired.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Because functions that are represented by power series are differentiable inside of their intervals of convergence, it follows that such functions are also continuous over their intervals of convergence.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Because functions that are represented by power series are differentiable inside of their intervals of convergence, it follows that such functions are also continuous over their intervals of convergence. Because functions that are represented by power series are continuous over their intervals of convergence, it follows from the Fundamental Theorem of Calculus that such functions are integrable and have antiderivatives over their intervals of convergence.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Because functions that are represented by power series are differentiable inside of their intervals of convergence, it follows that such functions are also continuous over their intervals of convergence. Because functions that are represented by power series are continuous over their intervals of convergence, it follows from the Fundamental Theorem of Calculus that such functions are integrable and have antiderivatives over their intervals of convergence. As a result, all of calculus applies to power series!

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

1 every antiderivative of f over (x0 − R, x0 + R) has the form

• f (x) dx = C +

∞

• k=0

ck k + 1(x − x0)k+1 for some constant C.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). Then

1 every antiderivative of f over (x0 − R, x0 + R) has the form

• f (x) dx = C +

∞

• k=0

ck k + 1(x − x0)k+1 for some constant C.

2 Every antiderivative of f has a radius of convergence of R.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Since we already established the validity of term-by-term differentiation, it should be clear that term-by-term integration is viable as well. (Just work backwards!)

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Since we already established the validity of term-by-term differentiation, it should be clear that term-by-term integration is viable as well. (Just work backwards!) The Fundamental Theorem of Calculus leads to the following.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Since we already established the validity of term-by-term differentiation, it should be clear that term-by-term integration is viable as well. (Just work backwards!) The Fundamental Theorem of Calculus leads to the following. Theorem Suppose that a function f is represented by a power series about x = x0 that has a nonzero radius of convergence R, that is f (x) =

∞

• k=0

ck(x − x0)k (x0 − R < x < x0 + R). If x0 − R < a < b < x0 + R, then b

a

f (x) dx =

∞

• k=0

b

a

ck(x − x0)k dx

• .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Integration in Action

For instance, given that we know the Maclaurin series for cos x is

∞

• k=0

(−1)k (2k)! x2k we can find the Maclaurin series for all of its antiderivatives via term-by-term integratiation.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Integration in Action

For instance, given that we know the Maclaurin series for cos x is

∞

• k=0

(−1)k (2k)! x2k we can find the Maclaurin series for all of its antiderivatives via term-by-term integratiation. ∞

• k=0

(−1)k (2k)! x2k

• dx =

∞

• k=0

(−1)k (2k)! x2k+1 (2k + 1) + C =

∞

• k=0

(−1)k (2k + 1)!x2k+1 + C,

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Term-by-Term Integration in Action

For instance, given that we know the Maclaurin series for cos x is

∞

• k=0

(−1)k (2k)! x2k we can find the Maclaurin series for all of its antiderivatives via term-by-term integratiation. ∞

• k=0

(−1)k (2k)! x2k

• dx =

∞

• k=0

(−1)k (2k)! x2k+1 (2k + 1) + C =

∞

• k=0

(−1)k (2k + 1)!x2k+1 + C, which we recognize as the Maclaurin series for sin x + C.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

We mentioned in the past that, except under special circumstances, infinite sums are generally not commutative nor associative as freely rearranging the terms of a series may alter the limiting behavior of the sequence of partial sums.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

We mentioned in the past that, except under special circumstances, infinite sums are generally not commutative nor associative as freely rearranging the terms of a series may alter the limiting behavior of the sequence of partial sums. We can make this assertion more precise.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

We mentioned in the past that, except under special circumstances, infinite sums are generally not commutative nor associative as freely rearranging the terms of a series may alter the limiting behavior of the sequence of partial sums. We can make this assertion more precise.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

We mentioned in the past that, except under special circumstances, infinite sums are generally not commutative nor associative as freely rearranging the terms of a series may alter the limiting behavior of the sequence of partial sums. We can make this assertion more precise. Theorem (Riemann Rearrangement Theorem) Rearranging the terms of an absolutely convergent series does not change the sum of the series. However, given a conditionally convergent series, the terms can be rearranged to converge to any real number M, diverge to an infinite limit, or fail to approach any limit, finite or infinite.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Classic Example: Grandi’s Series

Q: Where is the mistake in the following “equation”? 0 = 0 + 0 + 0 + · · · = (1 − 1) + (1 − 1) + (1 − 1) + · · · = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + · · · = 1 + 0 + 0 + 0 + · · · = 1

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Classic Example: Grandi’s Series

Q: Where is the mistake in the following “equation”? 0 = 0 + 0 + 0 + · · · = (1 − 1) + (1 − 1) + (1 − 1) + · · · = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + · · · = 1 + 0 + 0 + 0 + · · · = 1

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Classic Example: Grandi’s Series

Q: Where is the mistake in the following “equation”? 0 = 0 + 0 + 0 + · · · = (1 − 1) + (1 − 1) + (1 − 1) + · · · = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + · · · = 1 + 0 + 0 + 0 + · · · = 1 Therefore 0 = 1.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

A Classic Example: Grandi’s Series

Q: Where is the mistake in the following “equation”? 0 = 0 + 0 + 0 + · · · = (1 − 1) + (1 − 1) + (1 − 1) + · · · = 1 + (−1 + 1) + (−1 + 1) + (−1 + 1) + · · · = 1 + 0 + 0 + 0 + · · · = 1 Therefore 0 = 1. A: The series in lines 1-2 and the series in lines 3-4 are two completely different series, because they have completely different partial sums! (Remember that the sum of a series is defined to be the limit of the corresponding sequence of partial sums.) The third “equality” is thus spurious.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

The Alternating Harmonic Series Revisited

Consider the alternating harmonic series 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + · · · and rearrange the terms in the following fashion: 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + · · · .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

The Alternating Harmonic Series Revisited

Consider the alternating harmonic series 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + · · · and rearrange the terms in the following fashion: 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + · · · . This is indeed a rearrangement of the alternating harmonic series: every odd integer index occurs once positively, and the even integer indices occur once negatively (half of them as multiples of 4, the other half as twice odd integers).

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

The Alternating Harmonic Series Revisited

The terms of the rearrangement 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + · · ·

• ccur in blocks of three of the form

1 2k − 1− 1 2(2k − 1)− 1 4k = 1 2(2k − 1)− 1 4k , k = 1, 2, 3, . . .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

The Alternating Harmonic Series Revisited

The terms of the rearrangement 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + · · ·

• ccur in blocks of three of the form

1 2k − 1− 1 2(2k − 1)− 1 4k = 1 2(2k − 1)− 1 4k , k = 1, 2, 3, . . . The above series can therefore be written as 1 2−1 4+1 6−1 8+ 1 10− 1 12+· · · = 1 2

• 1 − 1

2 + 1 3 − 1 4 + 1 5 − 1 6 + · · ·

• Sections 9.5, 10.3: Finding and Using Taylor Series, part 1

Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

The Alternating Harmonic Series Revisited

The terms of the rearrangement 1 − 1 2 − 1 4 + 1 3 − 1 6 − 1 8 + 1 5 − 1 10 − 1 12 + · · ·

• ccur in blocks of three of the form

1 2k − 1− 1 2(2k − 1)− 1 4k = 1 2(2k − 1)− 1 4k , k = 1, 2, 3, . . . The above series can therefore be written as 1 2−1 4+1 6−1 8+ 1 10− 1 12+· · · = 1 2

• 1 − 1

2 + 1 3 − 1 4 + 1 5 − 1 6 + · · ·

• We have thus rearranged the terms of the alternating

harmonic series in such a way that we halved the sum of the

• riginal series!

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Absolute Convergence vs. Conditional Convergence Revisited

This is why we call them “conditionally” convergent series. They converge to a particular sum on the condition that the terms are added up in the order that they appear.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Absolute Convergence vs. Conditional Convergence Revisited

This is why we call them “conditionally” convergent series. They converge to a particular sum on the condition that the terms are added up in the order that they appear. This sort of thing does not happen with absolutely convergent series though!

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Absolute Convergence vs. Conditional Convergence Revisited

This is why we call them “conditionally” convergent series. They converge to a particular sum on the condition that the terms are added up in the order that they appear. This sort of thing does not happen with absolutely convergent series though! The terms of an absolute convergent series can be rearranged however you like! You will always get the same sum in the end.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Absolute Convergence vs. Conditional Convergence Revisited

This is why we call them “conditionally” convergent series. They converge to a particular sum on the condition that the terms are added up in the order that they appear. This sort of thing does not happen with absolutely convergent series though! The terms of an absolute convergent series can be rearranged however you like! You will always get the same sum in the end. This is yet another reason why we care whether or not a series converges absolutely.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

What Does This Mean for Power Series?

But remember: a power series converges absolutely over the interior of its interval of convergence.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

What Does This Mean for Power Series?

But remember: a power series converges absolutely over the interior of its interval of convergence. This implies that all of the standard arithmetic properties of finite sums do hold for power series.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

What Does This Mean for Power Series?

But remember: a power series converges absolutely over the interior of its interval of convergence. This implies that all of the standard arithmetic properties of finite sums do hold for power series. Power series thus follow all of the usual rules of algebra and calculus! This is a big deal, as we can now use basic algebra and calculus to expand our preexisting catalog of Taylor series with impunity!

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example: Maclaurin Series for x2 sin x

Find the Maclaurin series for x2 sin x.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example: Maclaurin Series for x2 sin x

Find the Maclaurin series for x2 sin x. Because we already know that the Maclaurin series for sin x is

∞

• k=0

(−1)k (2k + 1)!x2k+1 = x − x3 3! + x5 5! − x7 7! + · · · we take x2

∞

• k=0

(−1)k (2k + 1)!x2k+1 =

∞

• k=0

x2 · (−1)k (2k + 1)!x2k+1 =

∞

• k=0

(−1)k (2k + 1)!x2k+3 = x3 − x5 3! + x7 5! − x9 7! + · · ·

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example: Maclaurin Series for cos2x

Find the Maclaurin series for cos2x.

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II

More Examples of Calculating Radii and Intervals of Convergence Calculus with Power Series Algebra with Power Series

Example: Maclaurin Series for cos2x

Find the Maclaurin series for cos2x. Let us use the identity cos2x = 1 2(1 + cos(2x)) and the Maclaurin series for cos x to get 1 2

• 1 +

∞

• k=0

(−1)k (2k)! (2x)2k

• = 1

2

• 1 +

∞

• k=0

(−1)k22k (2k)! x2k

• = 1

2 +

∞

• k=0

(−1)k22k−1 (2k)! x2k = 1 − 2 2!x2 + 23 4! x4 − 25 6! x6 + · · · .

Sections 9.5, 10.3: Finding and Using Taylor Series, part 1 Calculus II