Limit of series with infinite product <mstyle displaystyle="true" scriptlevel="0"> <munder>

Corvattam9m87

Corvattam9m87

Answered question

2022-06-02

Limit of series with infinite product n 1 1 n k = 1 n ( 1 π k )

Answer & Explanation

Bruce Townsend

Bruce Townsend

Beginner2022-06-03Added 5 answers

I think it's convergent: For n 4 consider
k = 4 n ( 1 π k ) = k = 4 n k π k k = 4 n k 3 k = ( n 3 ) ! 4 5 n
= 6 ( n 3 ) ! n ! = 6 ( n 2 ) ( n 1 ) n .
Thus, for n 4
1 n k = 1 n | 1 π k | ( k = 1 3 | 1 π k | ) 6 ( n 2 ) ( n 1 ) n 2
and the right hand side in this inequality gives a convergent majorant.

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