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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?

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    $\begingroup$ 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. $\endgroup$ – John Omielan Jul 31 '19 at 3:00
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The last thing you said is correct : $$\lim_{n\to \infty} a_n \neq 0 \; \Rightarrow \; \sum_{n=1}^{\infty} a_n \textrm{ diverges.}$$

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Apply the divergence test.

$$\sum \frac{1 + (-1)^{n+1} (2n+1)}{4}$$ diverges since $$\frac{1 + (-1)^{n+1} (2n+1)}{4}$$ does not tend to zero as $n$ goes to $\infty$

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Hint: Compute the partial sums. Note that $a_n + a_{n+1}=0$ when $n$ is odd.

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