Determine if $\sum_{n=1}^\infty (-1)^n \frac{n+\sin(n)}{n+\cos(n)}$ is converging or diverging

Question

How can I determine if the series $$\sum_{n=1}^\infty (-1)^n \frac{n+\sin(n)}{n+\cos(n)}$$ is converging or diverging?

Answer 1

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.