Test the following series for convergence or divergence. Justify the decision.

First, we make the substitution , which gives us . Then we have

This integral we evaluate using integration by parts with

Therefore, we have

So, for the definite integral from to we have

But then the series is a telescoping series with

Hence,

Hence, the series converges.

You can also use the idea of areas from the previous question.

There is a simpler solution with the fact that .

Correction:

but then you have a difference of logs which is divergent, a better solution is e^-sqrt(x)x

but then you have a difference of logs which is divergent, a better solution is e^-sqrt(x) x