Question

if the series from 0 to infinity of cos 3 npi converges what is its sum

Answer 1

\[\sum_{n=0}^\infty \cos(3n\pi)~~\text{or}~~\sum_{n=0}^\infty \cos^3(n\pi)~~?\]
In either case, you can easily show that both series diverge. Use the \(n\)-th limit test for divergence.

Answer 2

That is, the test that says, given a series \(\sum a_n\), if \(\displaystyle\lim_{n\to\infty}a_n\not=0\), then \(\sum a_n\) diverges.