Prove that a series converges:

[mathkid2233]

Homework Statement

Let {a_n} be a sequence of positive numbers such that the sum of {a_n} from n = 1 to n = infinity converges. Show that the sum of {a_n} ^ (p / (p + 1)) from n = 1 to n = infinity also converges.

Homework Equations

n/a

The Attempt at a Solution

My professor provided me with some guidance here and I would like to double check and make sure that I'm on the right track. He told me to consider the terms with the sums with the property {a_n} ^ (1 / (p + 1)) <= 1/2. From this series, I believe that the series {a_n} ^ (p / (p + 1)) consisting of these terms would have to converge to 1/2. If all the terms were <= 1/2, then obviously this series would converge via the comparison test because {a_n} ^ (p / (p + 1) <= {a_n} ^ (p / (p +1)).

However, there is the case where {a_n} ^ (1 / (p + 1)) > 1/2. I changed the inequality so it now reads as 1 / ({a_n} ^ (1 / (p + 1))) < 2. Something should work out here but I don't see it at the moment.

 

[Dick]

That's not even true. Pick a_n=1/n^2. That converges. Pick p=1. (1/n^2)^(1/2)=1/n. That does not converge. Did you forget a premise?