Integral Method n n th Harmonic number for Sums B n = H n /2 - - PowerPoint PPT Presentation

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integralsum.1 Albert R Meyer, April 10, 2013

Mathematics for Computer Science MIT 6.042J/18.062J

Integral Method for Sums

integralsum.2 Albert R Meyer, April 10, 2013

Hn ::=1+ 1 2 + 1 3 +⋯+ 1 n

Harmonic Sums

nth Harmonic number Bn = Hn/2

integralsum.3 Albert R Meyer, April 10, 2013

1 x+1

0 1 2 3 4 5 6 7 8

1 1 2 1 3 1 2 1 1 3

Integral estimate for Hn

integralsum.4 Albert R Meyer, April 10, 2013

Integral estimate for Hn

Hn = area of rectangles > area under 1/(x+1) = 1 x +1 dx

n

∫

= 1 x

n+1

∫

dx = ln(n +1)

1

2

integralsum.7 Albert R Meyer, April 10, 2013

Book stacking

for overhang 3, need Bn ≥ 3 Hn ≥ 6 integral bound: ln(n+1) ≥ 6 so ok with n ≥  e6-1 = 403 books actually calculate Hn:

227 books are enough.

integralsum.8 Albert R Meyer, April 10, 2013

Book stacking

log(n+1)→∞ as n→∞, so overhang can be as big as desired!

integralsum.9 Albert R Meyer, April 10, 2013

CD cases over the edge 43 cases high --top 4 cases completely

• ff the table --1.8 or 1.9 case-lengths

integralsum.10 Albert R Meyer, April 10, 2013

stack of 43 CD’s

see 6.042 Spring02 demo page for credits

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integralsum.11 Albert R Meyer, April 10, 2013

don’t sneeze

int Albert R Meyer, April 10, 2013

0 1 2 3 4 5 6 7 8

1 1 2 1 3 1 2 1 1 3

1 x

egralsum.12

Upper bound for Hn

integralsum.13 Albert R Meyer, April 10, 2013

Integral Sum Bounds

f : ℝ+

+

→ ℝ Let be a weakly decreasing function.

n

S ::= ∑

n

f(i), I ::= ∫ f(x)dx

1 i=1

I + f(n) ≤ S ≤ I+ f(1)

integralsum.14 Albert R Meyer, April 10, 2013

Upper bound for Hn

 n 1  H <

n

∫ dx + 1  1 x  = 1 + ln(n)

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Asymptotic bound for Hn

ln(n+1) < Hn < 1+ ln(n)

Hn ∼ ln(n)

Albert R Meyer, April 10, 2013 integralsum.15

Asymptotic Equivalence

Def: f(n) ~ g(n)

f(n)   lim =1

n→∞ 

 g(n)  

Albert R Meyer, April 10, 2013 integralsum.16

Asymptotic Equivalence ~

Example: (n2 + n) ~ n2

pf:

2

n +n 1   lim

2

= lim 1+ =1

→∞ →∞

 n

n n

 

Albert R Meyer, April 10, 2013 integralsum.17

n

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