Proving Divergence of the Series: \sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}

[analysis001]

Homework Statement

Prove that the series diverges: \sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}

The Attempt at a Solution

I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}

Any clues on what to do next would be great. Thanks!

 

[Dick]

Homework Statement

Prove that the series diverges: \sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}

The Attempt at a Solution

I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}

Any clues on what to do next would be great. Thanks!

Try comparing with \frac{1}{\sqrt{n+1}+\sqrt{n+1}}. Does that converge or diverge? How is it related to your original series?

 

[Ray Vickson]

Homework Statement

Prove that the series diverges: \sum_{i=1}^{\infty}\sqrt{n+1}-\sqrt{n}

The Attempt at a Solution

I'm trying to use the comparison test, but I don't know what to compare it to. All I have done so far is change the terms to be in fraction form:
\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}

Any clues on what to do next would be great. Thanks!

Look at the behavior of $t_n \equiv \sqrt{n+1} - \sqrt{n}$ for large $n$. It helps to write
\sqrt{n+1} = \sqrt{n} \left( 1 + \frac{1}{n} \right)^{1/2}