# Sum of a power series - help!

1. Jan 2, 2013

### peripatein

1. The problem statement, all variables and given/known data
I am trying to find the sum of the series in the attachment.

2. Relevant equations

3. 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?

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2. Jan 2, 2013

### 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

3. Jan 2, 2013

### peripatein

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

4. Jan 2, 2013

### Dick

You are mistaken. Check it again.

5. Jan 2, 2013

### peripatein

Would you please explain how it was arrived at?

6. Jan 2, 2013

### Dick

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.