PlanetPhysics/Harmonic Series

The harmonic series $\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\dots$ satisfies the necessary condition of convergence $\underset{k\to \mathrm{\infty }}{\lim }{a}_n=0$ for the series \,$a_1+a_2+a_3+\dots$ of real or complex terms: $\underset{k\to \mathrm{\infty }}{\lim }\frac{1}{k}=0$ Nevertheless, the harmonic series diverges.\, It is seen if we first [[../TrivialGroupoid/|group]] the terms with parentheses: $1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+(\frac{1}{9}+\frac{1}{10}+\dots +\frac{1}{16})+\dots$ Here, each parenthetic sum contains a number of terms twice as many as the preceding one.\, The sum in the first parentheses is greater than\, $2\cdot \frac{1}{4}=\frac{1}{2}$,\, the sum in the second parentheses is greater than\, $4\cdot \frac{1}{8}=\frac{1}{2}$;\, thus one sees that the sum in all parentheses is greater than $\frac{1}{2}$.\, Consequently, the partial sum of $n$ first terms exceeds any given real number, when $n$ is sufficiently big.\\

The divergence of the harmonic series is very slow, though.\, Its speed may be illustrated by considering the difference $\sum _{k=1}^{n-1}\frac{1}{k}-\underset{1}{\overset{n}{\int }}\frac{dx}{x}=\sum _{k=1}^{n-1}\frac{1}{k}-\ln n$ (see the diagram).\, We know that $\ln n$ increases very slowly as $n\to \mathrm{\infty }$ (e.g. $\ln 1000000000\approx 20.7$).\, The increasing of the partial sum $\sum _{k=1}^{n-1}\frac{1}{k}$ is about the same, since the limit $\underset{n\to \mathrm{\infty }}{\lim }(\sum _{k=1}^{n-1}\frac{1}{k}-\ln n)=\gamma$ is a little positive number $\gamma =0.5772156649...$ which is called the Euler constant or Euler--Mascheroni constant .

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