Sum of a Power Series: Finding the Sum of a Series with a Variable

[toothpaste666]

Homework Statement

find the sum of the following series:

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

Homework Equations

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

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 :(

 

[micromass]

Do you know what

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

is?

 

[toothpaste666]

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

 

[micromass]

Now take derivatives.

 

[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.

 

[micromass]

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.

Yes, it is wrong. Please show your work.

 

[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
$-\frac{1}{(1-x)^2}$

 

[toothpaste666]

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

 

[micromass]

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

Indeed!

 

[toothpaste666]

thanks!