Homework Help: Power series

1. May 7, 2006

kezman

Find the sum of the series:
$$\sum\limits_{n = 1}^\infty {nx^n }$$ if $$\left| x \right| < 1$$

I thought maybe with the geometric form, but im not sure.

Last edited: May 7, 2006
2. May 7, 2006

Geekster

Is it asking for a number, or just if the series converges?

3. May 7, 2006

kezman

I think for a general solution. It should converge.

4. May 7, 2006

verd

Is it asking you to find a power series representation??

5. May 7, 2006

d_leet

It already is a power series... It's asking for an expression for the sum.

6. May 7, 2006

Geekster

You might start by writing out partial sums and see if that gets you anywhere.....

7. May 7, 2006

shmoe

How does this differ from your usual geometric series?

8. May 7, 2006

Curious3141

Hint : Call the original series S. Write out the first five or so terms in the series. Divide the series by x to get a new series (S/x). Now take the difference between this new series and the original series (S/x - S), term by term and see what you end up with.

The other way to do it is to differentiate a geometric series, but that's more complicated and unnecessary.

Last edited: May 7, 2006
9. May 8, 2006

Pyrrhus

Last edited by a moderator: Apr 22, 2017
10. May 8, 2006

Curious3141

Comparing series like these to derivatives of geometric series is a nice and interesting approach (I used to do this), but in most cases I've found that simply dividing or multiplying by x is an easier approach.

Last edited by a moderator: Apr 22, 2017
11. May 8, 2006

shmoe

May as well have a third approach:

$$\sum_{n=1}^{\infty}nx^n=\sum_{n=1}^{\infty}\sum_{i=1}^{n}x^n$$

Change the order of summation (absolutely convergent series) then apply geometric series a couple of times. This is maybe the most complicated of the three, practice in rearranging summations never hurt.

Last edited: May 8, 2006
12. May 8, 2006

kezman

thanks for all the hints.

The method I had to use is the derivative of the geometric series (similar to the one used for the maclaurin problem) using

$$\left( {\frac{1}{{1 - x}}} \right)^\prime = \sum\limits_{n = 0}^\infty {nx^{n - 1} }$$