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