Power Series

kingwinner

1) Evaluate
∞
∑ 1 / [n! (n+2)]
n=0
(hint: sum an appropriate power serires)


2) Evaluate
∞
∑ 1 / [n(n+1) 5^n]
n=1
(hint: start with a geometric series)


請高手賜教, thx!

Gnoehc

Consider
     
for the first one.

Consider
     

For the second one. Since two power serieses converge uniformly, termwise integration applies.

kingwinner

但問題係我唔知要點 integrate 先 work...

1) Integrating term by term, e^x = ∑ x^(n+1) / (n+1)!

2) Integrating term by term, ∑ x^(n+1) / (n+1)

但都唔係我想要既 series...

pls help...

[ 本帖最後由 kingwinner 於 10-4-7 06:34 AM 編輯 ]

Naozumi

你真係太死板啦

xe^x =  ∑ x^(n+1) / n!

跟住兩邊都求definite integral 咪得lor

kingwinner

thx, Q1 識做了, 但 Q2仲係唔識...

Q2)
∞
∑ x^n = 1/(1-x) for |x|<1
n=0

Take the indefinite integral of both sides...
∞
∑ x^(n+1) /(n+1) + C= -ln(1-x)
n=0
Put x=0 => C=0
∞
∑ x^(n+1) /(n+1) = -ln(1-x)
n=0

Divide both sides by x^2
∞
∑ x^(n-1) /(n+1) = - ln(1-x) / x^2
n=0

∞
∑ x^(n-1) /(n+1) = - [ln(1-x) / x^2] - 1/x
n=1


跟住點做?

thx

[ 本帖最後由 kingwinner 於 10-4-7 11:52 AM 編輯 ]

Gnoehc

It is not necessary to find the closed form of the integral (Though not hard)
     
If you are patient enough, you have finally (I just compute it through computer)