Determine if $\sum_{n=1}^\infty (-1)^n \frac{n+\sin(n)}{n+\cos(n)}$ is converging or diverging How can I determine if the series $$\sum_{n=1}^\infty (-1)^n \frac{n+\sin(n)}{n+\cos(n)}$$ is converging or diverging?
 A: You can usually test for convergence by trying to apply various convergence tests. Some tests may be inconclusive, but most problems about series that occur in a calculus course can be solved this way. First you make a list for yourself containing all convergence tests that were taught to you. Then, when you are faced with a problem, you go through them one by one and see which one does the trick. :-)
Hint for this particular problem:

 For the series $\sum_{n=1}^\infty x_n$ to converge it is necessary (though not generally sufficient) that the sequence $\{x_n\}_{n=1}^\infty$ converges to zero.

A: For the series to converge, as Josse said, the sequence being summed must itself tend to 0. Therefore I believe the following limits hold:
$$ \limsup_{x \to \infty} (-1)^x \frac {x+\sin x}{x+ \cos x}=1 \  and \ \liminf_{x \to \infty}(-1)^x \frac{x+\sin x}{x+ \cos x}=-1$$For discrete x. This implies that as n goes to infinity the series alternates between 1 and -1 and is therefore divergent. The limits themselves follow from the fact that as x goes to infinity, the sinx and cosx terms are negligible as they are stuck between 1 and -1, giving:
$$\lim_{x \to \infty} \frac{x+ \sin x}{x+\cos x}=1$$
Hope it helped. Plotting the graph of the function shows the alternating nature.
