Exercise Lover

March 10, 2007

Exercises A.1-8

Filed under: A.1 — yuhanlyu @ 2:08 am

\prod^n_{k=2} (1-1/k^2)
= \prod^n_{k=2} (k^2 - 1)/k^2
= \prod^n_{k=2} (k+1)(k-1)/k^2
= (n+1)/2n

Exercises A.1-7

Filed under: A.1 — yuhanlyu @ 2:02 am

\prod^n_{k=1} 2*4^k
= 2^k \prod^n_{k=1} 4^k
= 2^k * 4^{k(k+1)/2}
= 2^k * 2^{k(k+1)}
= 2^{k(k+2)}

Exercises A.1-6

Filed under: A.1 — yuhanlyu @ 1:57 am

Solution Wanted

Exercises A.1-5

Filed under: A.1 — yuhanlyu @ 1:51 am

Let f(x) = \sum^{\infty}_{k=0} x^k = 1/(1-x)
Let g(x) = f'(x) = \sum^{\infty}_{k=0} kx^{k-1} = 1/(1-x)^2
g(x) = \sum^{\infty}_{k=0} kx^{k-1}
= \sum^{\infty}_{k=1} (k+1)x^k
\sum^{\infty}_{k=1} (2k+1)x^{2k}
= (g(x) + g(-x))/2 when |x| < 1

Exercises A.1-4

Filed under: A.1 — yuhanlyu @ 1:39 am

\sum^{\infty}_{k=0} (k-1)/2^k
= -1 + \sum^{\infty}_{k=2} (k-1)/2^k
= -1 + (1/2)\sum^{\infty}_{k=1} k/2^k (change index k=2 to k=1)
= -1 + (1/2)\sum^{\infty}_{k=1} k/2^k
= -1 + (1/2)(1/2)/(1-(1/2))^2 (A.8 replace x by 1/2)
= 0

Exercises A.1-3

Filed under: A.1 — yuhanlyu @ 1:25 am

Let f(x) = \sum^{\infty}_{k=0} x^k = 1/(1-x)
f'(x) = \sum^{\infty}_{k=0} kx^{k-1} = 1/(1-x)^2
Let g(x) = xf'(x) = \sum^{\infty}_{k=0} kx^k = x/(1-x)^2
g'(x) = \sum^{\infty}_{k=0} k^2x^{k-1} = (x+1)/(1-x)^3
xg'(x) = \sum^{\infty}_{k=0} k^2x^k = x(x+1)/(1-x)^3

March 9, 2007

Exercises A.1-2

Filed under: A.1 — yuhanlyu @ 5:34 pm

\sum^n_{k=1} 1/(2k-1)
= \sum^{2n}_{k=1} 1/k - \sum^n_{k=1} 1/2k
= \ln 2n - (\ln n)/2 + O(1)
= \ln \sqrt{n} + O(1)

Exercises A.1-1

Filed under: A.1 — yuhanlyu @ 5:19 pm

\sum^n_{k=1} (2k-1)
= 2 \sum^n_{k=1}k - \sum^n_{k=1} 1
= n(n+2)

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