if Ʃa_n converges, with a_n > 0, then Ʃ(a_n)^2 always converges
The Attempt at a Solution
I am at a complete loss. I have tried using partial sums, cauchy criterion, and I tried using ratio test which seems to work but I am not sure.
Since Ʃa_n converges then by ratio test
lim n->∞ a_n+1 / a_n < 1
Now we apply ratio test to Ʃ(a_n)^2
lim n→∞ (a_n+1)^2 / (a_n)^2 = (lim n→∞ a_n+1 / a_n)^2 < 1^2 = 1
Thus by ratio test Ʃ(a_n)^2 converges.
This working did not utilize the condition a_n > 0, so it seems suspect to me.