Note that for each is and then besides the series converges by the integral test, thus the original series does converge.
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Of course your idea does work too. In order to prove that as we have
so it's easy to see that as and hence, the result.
My instinct is to use the root test, and try to show thatProve that
converges.
.
But unfortunately I don't know how to show this either.
Are there any tricks I can use on these nth root proofs which might help here?
Thanks!
Thank you very much, but I could still benefit from further assistance.
Once we have , the proof is easy for me. However, I am uncertain how you arrived at that inequality.
We have by definition
.
I have no idea how to prove that this series is increasing, but let's say for a moment that I could. In that case, we have, for each ,
.
Then algebraically we can show that
flipping:
.
Substitute back to get
,
and expand:
.
But this isn't good enough to get to our goal, which, again, is:
.
Apparently I need to learn more about exponents. Any more tricks would be much appreciated.