For which parameter this kind of series converges

Question

The sum is the following: $\sum_{k=1}^{\infty}\frac{a}{k^b}$

For which parameters this serie converges(a and b)?

Answer 1

One may apply the integral test to $\displaystyle f(x):=\frac{a}{x^b}$, $$ \int_1^N f(x)\,dx\le\sum_{k=1}^N f(k)\le f(1)+\int_1^N f(x)\,dx,\qquad N>1, $$ and see it converges iff $\,b>1$.