Prove that ∑n=1∞log⁡cos⁡(1n) converges absolutely.

FiessyFrimatsd0

FiessyFrimatsd0

Answered

2022-01-25

Prove that n=1logcos(1n) converges absolutely.

Answer & Explanation

spelkw

spelkw

Expert

2022-01-26Added 12 answers

As 0<cos1n<1, we have |logcos1n|=log(1cos1n)
Now cos1n=112n2+o(1n2), so
1cos1n=1112n2+o(1n2)=1+12n2+o(1n2)1+12n2
and ultimately
|logcos1n|log(1+12n2)12n2
which converges.

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