Finding the Sum of a Power Series: Tips and Tricks for Success

[peripatein]

Homework Statement

I am trying to find the sum of the series in the attachment.

Homework Equations

The Attempt at a Solution

I have tried to use various series and their derivatives, to not much avail.
I am not sure how to handle the n^2 factor.
Should I break it down to two series?
Any suggestions?

 

[lurflurf]

You know about derivatives? Good do this

$$\sum_{k=1}^\infty k^2 x^{n-1}=\left( x \left( x \sum_{k=1}^\infty x^{k-1}\right)^\prime \right)^\prime=\left( x \left( x \frac{1}{1-x}\right)^\prime \right)^\prime$$

|x|<1
your case will be x=1/10

 

[peripatein]

If I am not mistaken, this yields 700/729, which, according to Wolfram, is incorrect. Would you please account for that?

 

[Dick]

If I am not mistaken, this yields 700/729, which, according to Wolfram, is incorrect. Would you please account for that?

You are mistaken. Check it again.

 

[peripatein]

Would you please explain how it was arrived at?

 

[Dick]

Would you please explain how it was arrived at?

Basically you take x^(k-1). Multiplying by x and differentiating gives you k*x^(k-1). Doing the same thing again gives k^2*x^(k-1). Which is the form you want. Now sum the initial x^(k-1) as a geometric series and repeat the same sequence of operations on the function of you get.