convergence of $ \sum \frac{1+\cos(n)}{n+1}$ I was testing with some convergence series criteria, but I'm in trouble with the next:
$$ \sum \frac{1+\cos(n)}{n+1}$$
I am sure that direct comparison criteria works, but what is the correct series to compare?
 A: *

*$\sum \frac{\cos n}{n+1}$ converges by Dirichlet's test$\color{blue}{^{[1]}}$.

*$\sum \frac{1}{n+1}$ diverges by 
Integral test against $\int_0^n \frac{dx}{1+x}$.

*$\sum \frac{1+\cos n}{n+1}$ diverges because it is a sum of a convergent and a divergence sequence.
Notes
$\color{blue}{[1]}$ To apply the Dirichlet's Test, one need to verify two conditions


*

*$\frac{1}{n+1}$ is monotonic decreasing and converges to $0$. 

*The partials sums $\sum_{n=1}^N \cos n$ is bounded. 


The $1^{st}$ condition is obvious, the $2^{nd}$ condition is also true because
$$\sum_{n=1}^N \cos n = \Re \left[ \sum_{n=1}^N e^{in} \right] = \Re\left[ e^i \frac{1-e^{iN}}{1-e^i}\right]
\quad\implies\quad
\left|\sum_{n=1}^N \cos n\right| \le \frac{2}{| 1 - e^i|} = \frac{1}{\sin\frac12}
$$
If you don't like the use of complex number, you can also derive same inequality using trigonometry.
A: $\cos n\ge0$ often enough. The sequence $n$ hits at least one point of every interval $[(k-1/4)2\pi,(k+1/4)2\pi]$, $k\ge1$. This is enough to have the series diverge.
