Alternating series

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

An alternating series is an infinite series whose terms are alternately positive and negative: $$ u_1 - u_2 + \cdots + (-1)^{n-1}u_n + \cdots, \quad u_k > 0. $$ If the terms of an alternating series are monotone decreasing $(u_{n+1} < u_n)$ and tend to zero ($\lim_{n\rightarrow\infty} u_n = 0$), then the series is convergent (Leibniz' theorem). The remainder term of an alternating series, $$ r_n = (-1)^{n-1}u_n +\cdots, $$ has the same sign as its first term and is less then the latter in absolute value. The simplest examples of alternating series are $$ 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots + (-1)^{n-1} \frac{1}{n} + \cdots $$ and $$ 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + (-1)^{n-1} \frac{1}{2n-1} + \cdots. $$ The sum of the first of these series is $\log 2$; that of the second is $\pi/4$.