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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.