Does the series converge or diverge?

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

1 Answer
Oct 17, 2017

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.