# How to approach this series using comparison test: $\sum\limits_{n=1}^{\infty} \frac1{\sqrt{n^2-\frac12}}$?

I'm wondering if someone could give me a direction to go off of with the following series:

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

I'm wondering how I can show this series diverges/converges using the comparison test for series; this series kind of resembles $1/n$ which diverges, hence why I'm thinking something like this.

I'm honestly stuck, and would really appreciate a hint!

• You have $\sqrt{n^2 - \frac{1}{2}} \leq \sqrt{n^2}$. – Joe Johnson 126 May 17 '17 at 12:08

n is always > 1 and this allows us to use AM-GM inequality,

$$^3\sqrt\frac 1{(n-\frac 1{\sqrt{2}})(n+\frac 1{\sqrt{2}})} > \ ^3\sqrt{\frac{1}{n^2}}$$ (Which surprisingly is the same as using common sense; Nevermind)

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^2-\frac{1}{2}}} > \sum_1^\infty \frac{1}{n^{2/3}}$$

if n>1,

$$n^{2/3} Which is a consequence of the Trivial Inequality*,

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^2-\frac{1}{2}}} > \sum_1^\infty \frac{1}{n}$$

Which diverges.

There you go.

If you need more comfort with handling comparisons I suggest going through Inequalities like the Trivial Inequality, RMS-AM-GM-HM inequality and the Cauchy-Schwarz Inequality although these might not be that useful.

(*)$$n^2>n \rightarrow n^3>n^2 \rightarrow n>n^{2/3}$$ since we are taking principle roots and Real n>1