What is the nature of the series of the general term:
$u_n = \frac{e - (1 + \frac{1}{n})^n}{n^{\frac{3}{2}} - [n^{\frac{3}{2}}] + n} $.
[] is the step function.
Using the inequality of the step function I found:
$\frac{e - (1 + \frac{1}{n})^n}{n + 1} < u_n \leq \frac{e - (1 + \frac{1}{n})^n}{n} $
By computing the limit of the right and left terms I found: $\lim_{n \to + \infty} \frac{e - (1 + \frac{1}{n})^n}{n + 1} = \lim \frac{e - (1 + \frac{1}{n})^n}{n} = 0$ which is not helpful.
How can I find the nature of the series?