Why isn't this series convergent?

Question

Hello! I'm trying to solve the following series through the alternating series test:

cos(n pi) / n^(1/n), with the series starting at 1 and going to infinity.

I started by finding the first variables of the series, which is this

= [cos(pi)/1] + [cos(2pi)/2^(1/2)] + [cos(3pi)/3^(1/3)]

= [-1/1] + [1/2^(1/2)] + [-1/3^(1/3)]

= series (-1)^n / n^(1/n)

I isolated the bn a 1/(n^(1/n)), which converges to 0 as n goes to infinity. Due to this and the fact that bn is decreasing led me to the conclusion that the series is convergent. However, the study guide says that I'm incorrect.

I checked dezmos and it did show a steep curve to 0, and then it begins oscillating, which means I must be incorrect. How am I applying the alternating series test wrong to get this incorrect answer? Thank you for your time.

Answer 1

Hello Isabella,

Check the answer.