Absolute convergence of a series with integral inside

Question

Im having trouble solving the absolute convergence of this series. None of the common tests seem to work and so far couldn´t find any function to compare it to:

$\sum_{n=1}^\infty (-1)^{n} \int_{n}^{n+1}\frac{e^{-x}}{x}dx$

I would appreciate any suggestions.

Answer 1

You have$$\sum_{n=1}^\infty\int_n^{n+1}\frac{e^{-x}}x\,\mathrm dx=\int_1^\infty\frac{e^{-x}}x\,\mathrm dx<\int_1^\infty e^{-x}\,\mathrm dx$$and this last integral converges.