- #1

Rectifier

Gold Member

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**The problem**

In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## :

$$ \sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right) $$

**The attempt**

## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} = \frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}} ##

Now, I am supposed to somehow find:

## \lim_{k \rightarrow \infty} \frac{\frac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}}}{\frac{1}{k\sqrt{k}}} ##. I am not sure that this I have simplified the starting expression enough.

There are alternative solutions to this problem but I would like to solve it using limit comparison test. Thank you for your understanding.