Finding the Sum of a Convergent Power Series

[mathmathmad]

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?

 

[tiny-tim]

Hi mathmathmad!

(have an infinity: ∞ and try using the X² and X₂ tags just above the Reply box )

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?

Hint: try integrating.

 

[vintwc]

differentiating is a much better option fyi

 

[Mathnerdmo]

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).

 

[mathmathmad]

what to integrate?
intergrate n^2.x^n?

 

[tiny-tim]

what to integrate?
intergrate n^2.x^n?

Sort-of.

Suppose it was ∑ nxⁿ … what would you integrate?

 

[Mathnerdmo]

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.

 

[tiny-tim]

Hi Mathnerdmo!

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.

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