Cauchy criterion, harmonic series and the Riemann-zeta function

Let $(a_n{)}_{n=1}^{\mathrm{\infty }}$ be a decreasing sequence of non-negative real numbers. Then the series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ is convergent if and only if the series

is convergent. This is the Cauchy criterion.

Harmonic series

The Cauchy criterion can be used to show that the Harmonic series, [...], is [...]ergent. Yet [...] is [...]ergent.

Riemann-zeta function

The quantity $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{q^n}$ is called the Riemann-zeta function of q and is denoted by $\zeta (q)$. This function is very important in number theory in particular for investigating the distribution of primes. There is a famous unsolved problem regarding this function called the Riemann hypothesis.