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.