I am having a lot of trouble with these series questions. Up until this point, I had relatively little trouble with all the questions in the book. These seem to require knowledge about approximations of functions and other external experience-based knowledge, which I just don't have yet.
Determine convergence or divergence of the given series. In the case of convergence, determine whether the series converges absolutely or conditionally.
$\sum_{n=1}^\infty (-1)^n\left[e-\left(1+\frac 1 n \right)^n\right]$
It's easy to see that
$\lim_{n\to\infty}\left[e-\left(1+\frac 1 n\right)^n\right]=0$
however, in order to apply Leibniz's Rule and show conditional convergence I need to show that the sequence is monotonically decreasing. This doesn't seem doable with straight inequalities, so I tried taking the derivative, which just resulted in an uninterpretable mess. This doesn't even begin to address the question of absolute convergence/divergence.
There are 54 of these questions... I must be missing something really fundamental if they all take this long.