if $(a_n)$ be a sequence such that $a_i \geq a_j\;, \forall i < j$ and converges to 0, alternating series defined by $\displaystyle \sum\limits_{n} (-1)^{n+1}a_n$ converges. I went through the proof, and a similar argument can be said about $\displaystyle \sum\limits_{n} (-1)^{n}a_n$.
Basically, my question is, if the above condition regarding the sequence holds, can we say that the series $\displaystyle \sum\limits_{n} (-1)^{n}a_n$ also converges ?
If $$\displaystyle \sum\limits_{n} (-1)^{n}a_n$$ converges, then $$\displaystyle \sum\limits_{n} (-1)^{n+1}a_n$$ must converge to $$-\displaystyle \sum\limits_{n} (-1)^{n}a_n$$
Note that for the alternating convergent series we also need $$a_n \ge 0 $$