Test the following series for convergence or divergence. Justify the decision.
The series converges. To show this we apply the integral test (Theorem 10.11 on page 397 of Apostol), letting
As in the previous exercise we evaluate the integral using integration by parts with
Therefore,
But, from the previous exercise we know
converges; hence,
convergges.
Very nice proof!
I offer an alternative:
Since
the series
converges from T10.9 (below is the case for the limit being 0) since
is the Riemann Zeta function
which converges.
f(n) is not a decreasing function if you start from n=1. You need to start from n=3