the definition of this test is:
if $a_n$ decreases monotonically and goes to 0 in the limit then the alternating series $\sum_{n=1}^{\infty}(-1)^na_n$ converges
my question is: why does the series $a_n$ has to be monotonically descending,
isn't it enough for $\lim_{n\to\infty}a_n = 0$ ?
can someone give me an example for that?
Best Answer
Counter example:
Check the following alternating series diverges:
$$\sum_{n=1}^\infty(-1)^na_n\;\;,\;\;a_n=\begin{cases}0&,\;\;n\;\;\text{is odd}\\{}\\\frac1n&,\;\;n\;\;\text{is even}\end{cases}$$