Sum of a power series

1. Apr 20, 2014

toothpaste666

1. The problem statement, all variables and given/known data

find the sum of the following series:

$\sum_{n=1}^\infty nx^{n-1} , |x|<1$

2. Relevant equations

$\frac{a}{1-r}$

3. The attempt at a solution

i know that a function representation for that series is $-\frac{1}{(1-x)^2}$ but how is it possible to find the sum of a series with a variable in it? please help :(

Last edited: Apr 20, 2014
2. Apr 20, 2014

micromass

Staff Emeritus
Do you know what

$$\sum_{n=0}^{+\infty} x^n$$

is?

3. Apr 20, 2014

toothpaste666

$\frac{1}{1-x}$

4. Apr 20, 2014

micromass

Staff Emeritus
Now take derivatives.

5. Apr 20, 2014

toothpaste666

i know that the series as a function is $\frac{-1}{(1-x)^2}$ but webassign said that was wrong. they are looking for the sum of the series.

6. Apr 20, 2014

micromass

Staff Emeritus

7. Apr 20, 2014

toothpaste666

I wrote it as
$(1-x)^{-1 }$
to take the derivative i multiplied it by the exponent and subtracted one from the exponent.
$-1(1-x)^{-2}$
which is
$-\frac{1}{(1-x)^2}$

8. Apr 20, 2014

toothpaste666

oh wait i see it now. i forgot to use the chain rule. it should be
$\frac{1}{(1-x)^2}$

9. Apr 20, 2014

micromass

Staff Emeritus
Indeed!

10. Apr 21, 2014

thanks!