# Infinite series, does ∑ n/2^n diverge?

## Homework Statement

$$\sum^{\infty}_{n=1} \frac{n}{2^{n}}$$

Does this series converge or diverge?

## The Attempt at a Solution

By the Cauchy condensation test (http://en.wikipedia.org/wiki/Cauchy_condensation_test) I think this one diverges. But not sure if I am using it correctly.

According to the test,

$$\sum^{\infty}_{n=1} \frac{n}{2^{n}}$$

converges if and only if

$$\sum^{\infty}_{n=1} 2^{n} \frac{2^{n}}{2^{n}} = \sum^{\infty}_{n=1} 2^{n}$$

converges, which doesn't.

Thank you for any help.

Mark44
Mentor

## Homework Statement

$$\sum^{\infty}_{n=1} \frac{n}{2^{n}}$$

Does this series converge or diverge?

## The Attempt at a Solution

By the Cauchy condensation test (http://en.wikipedia.org/wiki/Cauchy_condensation_test) I think this one diverges. But not sure if I am using it correctly.

According to the test,

$$\sum^{\infty}_{n=1} \frac{n}{2^{n}}$$

converges if and only if

$$\sum^{\infty}_{n=1} 2^{n} \frac{2^{n}}{2^{n}} = \sum^{\infty}_{n=1} 2^{n}$$

converges, which doesn't.
In the line above, you aren't using the condensation test correctly. For your series, f(n) = n/(2n), so what would be f(2n)?

A test that would be simpler to apply would be the Limit Ratio Test.
Thank you for any help.

In the line above, you aren't using the condensation test correctly. For your series, f(n) = n/(2n), so what would be f(2n)?

A test that would be simpler to apply would be the Limit Ratio Test.

Ah, okay, I see it where I went wrong..

Using the limit ratio test

$$lim \left| a_{n+1}/a_{n} \right| = lim \left| \frac{(n+1)/2^{n+1}}{n/2^{n}} \right| = 1/2 < 1$$

So it converges...

Mark44
Mentor
Yes, indeed.