Math and science::Analysis::Tao::07. Series

# Cauchy criterion, harmonic series and the Riemann-zeta function

Let $$(a_n)_{n=1}^{\infty}$$ be a decreasing sequence of non-negative real numbers. Then the series $$\sum_{n=1}^{\infty}a_n$$ 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 $$\sum_{n=1}^{\infty}\frac{1}{q^n}$$ 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.