How Can You Find a Closed Form for the Series x + 2x² + 3x³ + 4x⁴ + ...?

[noblerare]

Homework Statement

I want to find a closed form formula for:

x+2x^2+3x^3+4x^4+\ldots

I know that this can be written as:

\sum_{n=1}^{\infty}nx^n

but I would like to have a closed formula for this.

The formula for an infinite geometric series is:
\sum_{n=0}^{\infty}x^n = \frac{1}{1-x}

Which is somewhat close but the series in question is not exactly geometric.

How do I go about doing this?

 

[rock.freak667]

Try differentiating the formula for the geometric series.

 

[Hurkyl]

Also, the usual "trick" for deriving geometric series also works for that one -- combine the original series S with the series xS to produce something simpler.

 

[rishicomplex]

this sort of series is called an arithmetic geometric progression (AGP) or something...like someone said, multiply by x and then subtract to get a simple geometric progression...in this case you could even divide by x