Does the series from 1 to infinity 1/(5n^2+1)^(1/3) converge or diverge?
I thought by using the divergence test and checking if the limit goes to 0 as n approaches infinity that the series converges. But the answer is "diverges." What test could I use to get divergence for this series?
I forced it into a comparison test . . .Converge or diverge? .
Add to both sides: .
Take the cube root of both sides: .
. . That is: .
Take the reciprocal of both sides: .
Summate both sides: .
We have: . . . . a divergent -series
Since is greater than a divergent series, diverges.
Okay, I'm looking over this again and I'm not sure I understand how this means divergence.
The original series is greater than or equal to the comparison series. I thought the comparison test only worked if the comparison series was greater than the original. And only worked then if the comparison series converges (and hence the original converges.) This is what I was taught in class anyway.