How Do You Solve This Arithmetico-Geometric Series?

[eNathan]

Trying to solve this infinite series??

Hey folks! I've spent hours trying to solve this and have exhausted all available resources.. I just need to be pointed in the right direction!

Homework Statement

Compute the sum of the infinite series (I believe this is an arithmetico geometric series):
\sum \frac{n+1}{4^{n}}

For n=0..infinity

Homework Equations

I know the standard way to solve a geometric series, but don't know how to solve this type of series.

The Attempt at a Solution

I've set up something like this:
S_{n} = \sum \frac{n+1}{4^{n}}
I've tried multiplying by 1/4, 4 and other logical things, but am just not sure how to proceed.

Thanks in advance!

 

[mfb]

Your expression can be written as
$$\sum_0^\infty \frac{1}{4^{n}} + \sum_1^\infty \frac{1}{4^{n}} + \sum_2^\infty \frac{1}{4^{n}} + \dots$$
(This assumes you start your sum at 0, if you start at 1 you have to modify it a bit.)

 

[haruspex]

Another approach, perhaps more general in application, is to multiply the terms by xⁿ then integrate. You should then be able to sum the series into closed form and differentiate to get a closed form for the original sum. Finally set x=1.