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

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)
     

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 ]

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

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 ]

Gnoehc

Consider
     
for the first one.

Consider
     

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