A sum that's possibly equal to the Euler-Mascheroni Constant <munderover> &#x2211;<!-- ∑ -->

fythynwyrk0

fythynwyrk0

Answered question

2022-07-10

A sum that's possibly equal to the Euler-Mascheroni Constant n = 1 ln n ! n 3

Answer & Explanation

Keely Fernandez

Keely Fernandez

Beginner2022-07-11Added 14 answers

It is known that for n 1
log n ! > ( n + 1 2 ) log n n + log 2 π + 1 12 n 1 360 n 3 .
log n ! > ( n + 1 2 ) log n n + log 2 π + 1 12 n 1 360 n 3 .
Thus
n = 1 log n ! n 3 > n = 1 log n n 2 + 1 2 n = 1 log n n 3 n = 1 1 n 2 + log 2 π n = 1 1 n 3 + 1 12 n = 1 1 n 4 1 360 n = 1 1 n 6 = ζ ( 2 ) 1 2 ζ ( 3 ) ζ ( 2 ) + ζ ( 3 ) log 2 π + 1 12 ζ ( 4 ) 1 360 ζ ( 6 ) = 0.583661 > γ .

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