# Checking the convergence of a requested series

Does the series $$\sum \frac{1 + (-1)^{n+1} (2n+1)}{4}$$ converge$$?$$

I partitioned the series in two parts, when n is even $$\sum -\frac{n}{2}$$ and when $$n$$ is odd, $$\sum \frac{n+1}{2}$$

Both series are divergent, so this doesn't conclude anything.

$$N^\text{th}$$ terms of these series are not zero. Can I say from here that the series is divergent or there's another way to check the convergence of such series?

• I assumed you didn't want to check the "convenience" of a series, so I fixed your title accordingly. Since the system didn't allow me to change just that one word, I added the "requested" adjective. – John Omielan Jul 31 '19 at 3:00

The last thing you said is correct : $$\lim_{n\to \infty} a_n \neq 0 \; \Rightarrow \; \sum_{n=1}^{\infty} a_n \textrm{ diverges.}$$
$$\sum \frac{1 + (-1)^{n+1} (2n+1)}{4}$$ diverges since $$\frac{1 + (-1)^{n+1} (2n+1)}{4}$$ does not tend to zero as $$n$$ goes to $$\infty$$
Hint: Compute the partial sums. Note that $$a_n + a_{n+1}=0$$ when $$n$$ is odd.