Convergence of a series of integrals: $S=sum_{ngeq 1}(-1)^{n}int_{n}^{n+1}frac{1}{t e^t},dt$

Set $$b_n=int_{n}^{n+1}frac{1}{t e^t},dt,$$ then $$frac{1}{(n+1)e^{n+1}}leq b_nleqfrac{1}{ne^{n}},$$ and using squeeze theorem, $$lim_{nto +infty}{b_n}=0.$$

Obviously, ${b_n}$ is decreasing, therefore $S=sum_n{(-1)^nb_n}$ is convergent.

The function begin{equation} f(t)=frac{1}{te^t} end{equation} is continuous, strictly decreasing on $[1,+infty)$ and its limit as $tto+infty$ is $0$. This is enough to conclude that the terms begin{equation} a_n=int_n^{n+1}frac{1}{te^t},text{d}t end{equation} are decreasing and converge to $0$ as $nto+infty$. Indeed, by the mean value theorem, there are $t_nin[n,n+1]$ and $t_{n+1}in[n+1,n+2]$ such that $a_n=f(t_n)$ and $a_{n+1}=f(t_{n+1})$, thus $a_n=f(t_n)>f(t_{n+1})=a_{n+1}$ and $a_n=f(t_n)to 0$ as $nto+infty$. All the requirements of the alternating series test are therefore fulfilled and allow to conclude that the series converges.