sequences and series - Convergence of $sum_{n=0}^{infty}(-1)^n frac{2+(-1)^n}{n+1}$

I have to show that the following series convergences:

$$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$

I have tried the following:

• The alternating series test cannot be applied, since $\frac{2+(-1)^n}{n+1}$ is not monotonically decreasing.


• I tried splitting up the series in to series $\sum_{n=0}^{\infty}a_n = \sum_{n=0}^{\infty}(-1)^n \frac{2}{n+1}$ and $\sum_{n=0}^{\infty}b_n=\sum_{n=0}^{\infty}(-1)^n \frac{(-1)^n}{n+1}$. I proofed the convergence of the first series using the alternating series test, but then i realized that the second series is divergent.


• I also tried using the ratio test: for even $n$ the sequence converges to $\frac{1}{3}$, but for odd $n$ the sequence converges to $3$. Therefore the ratio is also not successful.

I ran out of ideas to show the convergence of the series.

Thanks in advance for any help!

Answer

It is not convergent. To see this, let
$$
a_n = (-1)^n\frac{2}{n+1},\qquad b_n =\frac{1}{n+1},\qquad c_n = a_n + b_n.
$$
The series $\sum a_n$ is convergent by the alternating test.

We are interested in the convergence of $\sum c_n$. If $\sum c_n$ was convergent, then $\sum b_n = \sum c_n - \sum a_n$ would also be convergent, which is known to be false (divergence of the harmonic series).