PlanetPhysics/Harmonic Series

The harmonic series $$\sum_{k=1}^\infty\frac{1}{k} \;=\; 1+\frac{1}{2}+\frac{1}{3}+\ldots$$ satisfies the necessary condition of convergence $$\lim_{k\to\infty}a_n \;=\; 0$$ for the series \,$$a_1+a_2+a_3+\ldots$$ of real or complex terms: $$\lim_{k\to\infty}\frac{1}{k} \;=\; 0$$ Nevertheless, the harmonic series diverges.\, It is seen if we first group the terms with parentheses: $$1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right) +\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right) +\left(\frac{1}{9}+\frac{1}{10}+\ldots+\frac{1}{16}\right)+\ldots$$ 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 $$\sum_{k=1}^{n-1}\frac{1}{k}-\!\int_1^n\frac{dx}{x} \;=\; \sum_{k=1}^{n-1}\frac{1}{k}-\ln{n}$$ (see the diagram).\, We know that $$\ln{n}$$ increases very slowly as $$n \to \infty$$ (e.g. $$\ln{1\,000\,000\,000} \,\approx\, 20.7$$).\, The increasing of the partial sum $$\sum_{k=1}^{n-1}\frac{1}{k}$$ is about the same, since the limit $$\lim_{n\to\infty}\left(\sum_{k=1}^{n-1}\frac{1}{k}-\ln{n}\right)\;=\;\gamma$$ is a little positive number $$\gamma \;=\; 0.5772156649...$$ which is called the Euler constant or Euler--Mascheroni constant.

\begin{pspicture}(-0.5,-0.5)(6,6) \psaxes[Dx=1,Dy=1]{->}(0,0)(-0.5,-0.5)(5.5,5.5) \rput(5.5,-0.2){$$x$$} \rput(-0.2,5.5){$$y$$} \rput(-0.3,-0.3){$$0$$} \psplot[linecolor=red]{0.22}{5.2}{1 x div} \psline(1,0)(1,1) \psline[linecolor=blue](1,1)(2,1)(2,0.5)(3,0.5)(3,0.333)(4,0.333)(4,0.25)(5,0.25)(5,0.2) \psline(2,0)(2,0.5) \psline(3,0)(3,0.333) \psline(4,0)(4,0.25) \rput(1,3.5){$$y = \frac{1}{x}$$} \end{pspicture}