Question

Does the series converge or diverge?

`sum_(n=1)^oo e^(1/n)/n`

Answer 1

The series would diverge.

Explanation:

nth term of the series can be written as `1/n e^(1/n)` and then express `e^(1/n)` as a power series `1+ 1/n +1/n^2 1/(2!) +.....`. The series would then become `sum_(n=1)^(oo)` `1/n` + `sum_(n=1)^oo (1/n^(2) + 1/n^(3) *1/(2!) +.....)`

The first series `sum_(n=1)^(oo)` `1/n` is a harmonic series, hence divergent, while all other series will converge because of p-series test.

In aggregate the given series would thus diverge.