# Homework Help: Help on infinite series

1. Sep 5, 2010

### noblerare

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

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?

2. Sep 5, 2010

### rock.freak667

Try differentiating the formula for the geometric series.

3. Sep 5, 2010

### Hurkyl

Staff Emeritus
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.

4. Sep 6, 2010

### 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