Finding the sum of an infinite series

[jspectral]

Homework Statement

$$\sum\frac{1}{n2^(n+1)}$$ from 1 to infinity.

By the way, that 2 is to the power of (n+1), doesn't show clearly.

Homework Equations

The Attempt at a Solution

I have worked out the first few individual calculations, up to n=6, and i know it approaches ln(2)/2, however I have no idea how to actually prove this.

 

[jbunniii]

What sorts of theorems do you know about power series?

 

[jspectral]

What sorts of theorems do you know about power series?

A few, it's for a 2nd year university subject. We've been using proof by induction a fair bit, but other than that I don't know the actual names for them.

 

[TheFurryGoat]

What do you know of integrating power series?

 

[jspectral]

What do you know of integrating power series?

You mean int(1 + x + x^2 + x^3 + ...) = x + x^2/2 + x^3/3 + ... ?

 

[jbunniii]

You mean int(1 + x + x^2 + x^3 + ...) = x + x^2/2 + x^3/3 + ... ?

Yes. See if you can work out how to apply that property to your problem.

(Hint: start by replacing the "1/2" with a variable.)

By the way, you can click on the following equation if you want to see how to typeset the sum properly:

$$\sum_{n=1}^{\infty}\frac{1}{n 2^{n+1}}$$