Harmonic series

The series of numbers \begin{equation} \sum_{k=1}^{\infty}\frac{1}{k}. \end{equation} Each term of the harmonic series (beginning with the second) is the harmonic mean of its two contiguous terms (hence the name harmonic series). The harmonic series is divergent (G. Leibniz, 1673), and its partial sums \begin{equation} S_n = \sum_{k=1}^n\frac{1}{k} \end{equation} increase as $\ln n$ (L. Euler, 1740). There exists a constant $\gamma>0$, known as the Euler constant, such that $S_n = \ln n + \gamma + \varepsilon_n$, where $\lim\limits_{n\to\infty}\varepsilon_n = 0$. The series \begin{equation} \sum_{k=1}^{\infty}\frac{1}{k^{\alpha}} \end{equation} is called the generalized harmonic series; it is convergent for $\alpha>1$ and divergent for $\alpha\leq1$.

Comments

For a proof of the expression for $S_n$ see, e.g., [a1], Thm. 422. Note that the series $\sum 1/p$ extended over all prime numbers $p$ diverges also; see, e.g., [a1], Thm. 427, for an expression of its partial sums.

Generalized harmonic series are often used to test whether a given series is convergent or divergent by estimating in terms of $1/n^{\alpha}$ the order of the terms of the given series; see Series.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)