\( \newcommand{\matr}[1] {\mathbf{#1}} \newcommand{\vertbar} {\rule[-1ex]{0.5pt}{2.5ex}} \newcommand{\horzbar} {\rule[.5ex]{2.5ex}{0.5pt}} \)
header
Show Answer
\( \newcommand{\cat}[1] {\mathrm{#1}} \newcommand{\catobj}[1] {\operatorname{Obj}(\mathrm{#1})} \newcommand{\cathom}[1] {\operatorname{Hom}_{\cat{#1}}} \)
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.