Convergence of riemann zeta function when s=3

Writing the infinite series as a limit of a partial sum, you can see that the series represents the left-endpoint Riemann approximation to the function :



In particular, since is a concave function for , i.e. strictly decreasing and approaching 0 as , which means the Riemann sum approximation is greater than the value of the integral.







So the value of the series is some real number smaller than 2 (actually somewhere around 1.202).

Long story short: use the integral test for determining whether the series converges.

A. is the answer in this question