posted by Bob .
The sum from n=1 to infinity of cos(npi/3)/n!
Does this absolutely converge, conditionally converge, or diverge?
Group every 6 terms (1 cycle) together and consider as an aggregate term for a new series, ΣQn. Assume that all the terms are positive because of the n! in the denominator.
If you can prove that
Qn+1/Qn < 1, then
by the ratio test, the (new) series is absolutely convergent.
Can you take it from here?