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.
There is a simpler solution with the fact that
.
Correction: