Is the convererse true for the alternating series test?
For example if $\displaystyle b_{n+1} \leq b_n$ is not true and $\displaystyle \lim_{n\to0} b_n=0$ is not true, does that mean the series diverges?
Is the convererse true for the alternating series test?
For example if $\displaystyle b_{n+1} \leq b_n$ is not true and $\displaystyle \lim_{n\to0} b_n=0$ is not true, does that mean the series diverges?
if $\displaystyle b_{n+1} \leq b_n$ is not true but $\displaystyle \lim_{n\to0} b_n=0$ is true then the series may converge.
Consider:
$\displaystyle S=\sum_{n=0}^{\infty} a_n$
$\displaystyle a_n = \begin{cases} (1/3)^n & n \text{ even} \\ -(1/2)^{n-1} & n \text{ odd} \end{cases}$
CB