- #1

- 1,086

- 2

## Homework Statement

Show [itex]\displaystyle\sum\limits_{n=0}^\infty (\frac{1}{2})^{\sqrt{n}}[/itex] converges.

## The Attempt at a Solution

I've pretty much tried all convergence tests and failed miserably. Both the ratio and the root test are inconclusive, as the limits are 1. I can't really use the integral test, and I don't know what else to compare this to to show that the series indeed converges. I know it does, but I just can't show it.

One idea I had is to rewrite the sum as [itex]\displaystyle\sum\limits_{n=0}^\infty (\frac{1}{2})^{\sqrt{n}} = \displaystyle\sum\limits_{n=0}^\infty (\frac{1}{2^{\frac{1}{\sqrt{n}}}})^{n}[/itex], where the thing in brackets is less than 1 for all n. Knowing the geometric series converges for r < 1, this would imply this series also converges... Except that I don't think reasoning is any good, since for any such r < 1, the thing in brackets will eventually get bigger than r. This is all I've got now, though.

Ugh, any help here would be greatly appreciated, as always.