Alternating series theorem fails if signs are not strictly alternating

Statement

Suppose we have a series of the form:

$a_{1}+a_{2}+\dots +a_{n}+\dots =\sum _{k=1}^{\infty }a_{k}$

such that the following conditions hold:

1. Infinite sign switching: All the $a_{k}$s are nonzero and the sign of $a_{k}$ switches infinitely often, i.e., there are infinitely many positive and infinitely many negative values of the $a_{k}$s.
2. Monotonically decreasing in magnitude: $|a_{k}|\geq |a_{k+1}|$ for all $k$.
3. Terms approach zero: $\lim _{k\to \infty }a_{k}=0$.

Then, any of the following is possible for the series:

• The series is an absolutely convergent series
• The series is a conditionally convergent series
• The series diverges to $+\infty$ or $-\infty$
• The partial sums of the series have differing values of limit superior and limit inferior.

Proof

Case	Example of series
absolutely convergent series	$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k^{2}}}=1-{\frac {1}{4}}+{\frac {1}{9}}-{\frac {1}{16}}+{\frac {1}{25}}\dots$
conditionally convergent series	$\sum _{k=1}^{\infty }{\frac {(-1)^{k}}{k}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}\dots$
divergent series	$1+{\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+{\frac {1}{7}}+{\frac {1}{8}}-{\frac {1}{9}}\dots$