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Evaluation of power series

  • #1

Homework Statement



evaluate ∑ n^2.x^n where 0<x<1

Homework Equations





The Attempt at a Solution


let a_n = n^2 and c=0
then radius of convergence, R=1
hence the series convergences when |x|<1
let f(x) = ∑ n^2.x^n
then f'(x) = ∑ n^3.x^n-1 for n=0 to infinity
then f'(x) = ∑ (n+1)^3.x^n for n=1 to infinity

from here, how to I derive a function ∑ (n+1)^3.x^n so as to integrate it to get the sum?
 

Answers and Replies

  • #2
tiny-tim
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Hi mathmathmad! :smile:

(have an infinity: ∞ and try using the X2 and X2 tags just above the Reply box :wink:)
let f(x) = ∑ n^2.x^n
then f'(x) = ∑ n^3.x^n-1 for n=0 to infinity
then f'(x) = ∑ (n+1)^3.x^n for n=1 to infinity …
Why are you making it more complicated? :redface:

Hint: try integrating. :wink:
 
  • #3
24
0
differentiating is a much better option fyi
 
  • #4
41
0
It's somewhat hard to start from your f(x) and find a series you know.

Instead, try starting from a series you know and apply these methods to get f(x).
 
  • #5
what to integrate?
intergrate n^2.x^n?
 
  • #6
tiny-tim
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what to integrate?
intergrate n^2.x^n?
Sort-of.

Suppose it was ∑ nxn … what would you integrate? :wink:
 
  • #7
41
0
I still say (expanding a little on my hint) to start with an expression for

∑ xn

and try to derive an expression for your series.
 
  • #8
tiny-tim
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Hi Mathnerdmo! :smile:
It's somewhat hard to start from your f(x) and find a series you know.

Instead, try starting from a series you know and apply these methods to get f(x).
I still say (expanding a little on my hint) to start with an expression for

∑ xn

and try to derive an expression for your series.
ah, i see what you mean now …

your method is basically the same as mine, but in reverse …

i'm integrating the question to try to get something easier, while you're starting with something easier, and differentiating to get the question. :wink:

Yes, if mathmathmad wants to start with ∑ xn and differentiate it, that's fine. :smile:
 

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