Does ∑n=1∞1n1+1n converge? If yes, to where?

Shamar Padilla

Shamar Padilla

Answered

2022-01-22

Does
n=11n1+1n
converge? If yes, to where?

Answer & Explanation

saennwegoyk

saennwegoyk

Expert

2022-01-23Added 7 answers

The series does not converge.
Consider whether n1nk for some constant k. If k=e, we have
1nlnn1lnnn
But for all n1,lnn<n, so we have an upper bound on n1n which means that the original sum is asymptotic to n=11n
mihady54

mihady54

Expert

2022-01-24Added 13 answers

You can try the limit comparison theorem (more general form):
1nnn1nlogn=lognnn
and then your series diverges since the other diverges.
For the other one: you can use Cauchys
RizerMix

RizerMix

Expert

2022-01-27Added 437 answers

2nn for n1 is easy to show. Hence 2n1/n, so 1n1+1/n12n which implies 1n1+1/n121n, which diverges.

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