Power Series (Overview) Bernd Schr oder logo1 Bernd Schr oder - - PowerPoint PPT Presentation
Summary Radius of Convergence Representation of Functions Power Series (Overview) Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Power Series (Overview) Summary Radius of
logo1 Summary Radius of Convergence Representation of Functions
Bernd Schr¨
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
n=0 is a sequence of numbers, then ∞
n=0
cn(x−x0)n is called a power series about x0.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
n=0 is a sequence of numbers, then ∞
n=0
cn(x−x0)n is called a power series about x0.
which could be infinity, so that the power series converges for every x in (x0 −R,x0 +R) and so that the power series diverges for every x not in (x0 −R,x0 +R).
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
n=0 is a sequence of numbers, then ∞
n=0
cn(x−x0)n is called a power series about x0.
which could be infinity, so that the power series converges for every x in (x0 −R,x0 +R) and so that the power series diverges for every x not in (x0 −R,x0 +R).
x0 +R, further convergence tests for series are needed.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
n=0 is a sequence of numbers, then ∞
n=0
cn(x−x0)n is called a power series about x0.
which could be infinity, so that the power series converges for every x in (x0 −R,x0 +R) and so that the power series diverges for every x not in (x0 −R,x0 +R).
x0 +R, further convergence tests for series are needed.
many series: Apply the ratio test. The power series converges for all x for which lim
n→∞
cnxn
diverges for all x for which the limit is greater than 1.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x| = lim
n→∞2n+1
n+2|x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x| = lim
n→∞2n+1
n+2|x| = 2|x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x| = lim
n→∞2n+1
n+2|x| = 2|x|
!
< 1,
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x| = lim
n→∞2n+1
n+2|x| = 2|x|
!
< 1, which is the case exactly when |x| < 1 2.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)+1xn+1 (−2)n n+1 xn
lim
n→∞
n+2 xn+1 2n n+1xn
n→∞
2n+1 n+2 n+1 2n |x| = lim
n→∞2n+1
n+2|x| = 2|x|
!
< 1, which is the case exactly when |x| < 1
2.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x| = lim
n→∞
1 n+1|x|
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x| = lim
n→∞
1 n+1|x| =
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x| = lim
n→∞
1 n+1|x| =
!
< 1,
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x| = lim
n→∞
1 n+1|x| =
!
< 1, which is always true.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
lim
n→∞
(n+1)!xn+1 1 n!xn
lim
n→∞
n! (n+1)!|x| = lim
n→∞
n·(n−1)···3·2·1 (n+1)·n·(n−1)···3·2·1|x| = lim
n→∞
1 n+1|x| =
!
< 1, which is always true. Hence R = ∞.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
is has the same derivatives as the function f at x0.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
is has the same derivatives as the function f at x0.
equal to the function on some interval.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
is has the same derivatives as the function f at x0.
equal to the function on some interval. (Warning: there are pathological functions whose Taylor series converge everywhere, but not to the function, except at x0.)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
is has the same derivatives as the function f at x0.
equal to the function on some interval. (Warning: there are pathological functions whose Taylor series converge everywhere, but not to the function, except at x0.)
are also called McLaurin series.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
about x0 is Tf (x) =
∞
n=0
f (n)(x0) n! (x−x0)n.
is has the same derivatives as the function f at x0.
equal to the function on some interval. (Warning: there are pathological functions whose Taylor series converge everywhere, but not to the function, except at x0.)
are also called McLaurin series.
∞
n=0
1 n!xn.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero. So we only need to focus on exponents of the form 2n+1, that is, we will have summands that involve x2n+1 (2n+1)!.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero. So we only need to focus on exponents of the form 2n+1, that is, we will have summands that involve x2n+1 (2n+1)!. The odd numbered derivatives at 0 are 1 and −1 alternatingly.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero. So we only need to focus on exponents of the form 2n+1, that is, we will have summands that involve x2n+1 (2n+1)!. The odd numbered derivatives at 0 are 1 and −1 alternatingly. Alternating negative signs are represented by (−1)n or (−1)n+1.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero. So we only need to focus on exponents of the form 2n+1, that is, we will have summands that involve x2n+1 (2n+1)!. The odd numbered derivatives at 0 are 1 and −1 alternatingly. Alternating negative signs are represented by (−1)n or (−1)n+1. For this one, (−1)n works.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Compute the Taylor Series of f(x) = sin(x) About x0 = 0 f(x) = sin(x), f(0) = 0 f ′(x) = cos(x), f ′(0) = 1 f ′′(x) = −sin(x), f ′′(0) = 0 f ′′′(x) = −cos(x), f ′′′(0) = −1 f ′′′′(x) = sin(x), f ′′′′(0) = 0 The pattern repeats. Even numbered derivatives are zero. So we only need to focus on exponents of the form 2n+1, that is, we will have summands that involve x2n+1 (2n+1)!. The odd numbered derivatives at 0 are 1 and −1 alternatingly. Alternating negative signs are represented by (−1)n or (−1)n+1. For this one, (−1)n works. So the Taylor series
∞
n=0
(−1)n (2n+1)!x2n+1.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
1 1+x2
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
1 1+x2 = 1 1−(−x2)
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
1 1+x2 = 1 1−(−x2) =
∞
n=0
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)
logo1 Summary Radius of Convergence Representation of Functions
∞
n=0
xn = 1 1−x for |x| < 1.
1 1+x2 = 1 1−(−x2) =
∞
n=0
=
∞
n=0
(−1)nx2n.
Bernd Schr¨
Louisiana Tech University, College of Engineering and Science Power Series (Overview)