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Revision difference : harmonic series
Version 2 - Version 1
1c1 < xxxxxxxx --- > yyyyyyyy 13c13,23 < Here, each parenthetic sum contains a number of terms twice as many as the preceding one.\, The sum in the first pair of parentheses is graater than\, $2\cdot\frac{1}{4} = \frac{1}{2}$,\, the sum in the second pair of parentheses is greater than\, $4\cdot\frac{1}{8} = \frac{1}{2}$;\, thus one sees that the sum in all pairs of 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. --- > Here, each parenthetic sum contains a number of terms twice as many as the preceding one.\, The sum in the first pair of parentheses is graater than\, $2\cdot\frac{1}{4} = \frac{1}{2}$,\, the sum in the second pair of parentheses is greater than\, $4\cdot\frac{1}{8} = \frac{1}{2}$;\, thus one sees that the sum in all pairs of 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 \PMlinkescapetext{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 \emph{Euler constant} or \emph{Euler--Mascheroni constant}.
