View source for Alternating series

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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 $\pi/4$.