Does the following series converge? \sum_{n=1}^\infty\frac{1}{n\root(n)(n)}

Rylan Duncan

Rylan Duncan

Answered question

2022-01-24

Does the following series converge?
n=11nnn

Answer & Explanation

tainiaadjouctlw

tainiaadjouctlw

Beginner2022-01-25Added 14 answers

Note that nn2. This can be proved by induction, for it is equivalent to n2n.
Thus
1nnn12n
It follows by Comparison with (half of) the harmonic series that our series diverges.
Lily Thornton

Lily Thornton

Beginner2022-01-26Added 19 answers

Limit comparison test:
1nnn1n=1nn1
So that both
n=11nnn and n=11n
converge or both diverge...
RizerMix

RizerMix

Expert2022-01-27Added 656 answers

AM-GM gives
S=123120.2937
1nnn>121n
son=1M1nnn>12(n=1M12)

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