Do you recognize the indeterminate form,
Some one help please. Have been struggling with this for more than an hour..
sum (n/(n+1))^(n^2), where n goes from 1 to infinity.
How to show that this series converge when n goes from 1 to infinity?
I try root test but L goes to one, which mean the test is inconclusive (thus ratio test doesn't work, too).
Thanks so much!
Consider the logarithm of the general term.
Use log rules to get the n^2 out of the exponent.
Rewrite the n^2 as an awkward reciprocal in the denominator.
Think about this as an Indeterminate Form that you are used to seeing.
Use log rules to simplify the numerator
Apply l'Hospital.
See if we are getting anywhere.
Well, I suggested putting 1/n^2 in the denominator, but this will do for the demonstration.
It's not "1" in the denominator. log(1) = 0 ==> And the reciprocal of that would be...
Now, put it back in the numerator and put the 1/n^2 in the denominator. Trust me on this. Your life will be much easier.