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enrotlavaec

enrotlavaec

Answered question

2022-06-16

Suppose n = 0 a ( n ) < , what about n = 2 a ( n log ( n ) ) ?

Answer & Explanation

marktje28

marktje28

Beginner2022-06-17Added 22 answers

Suppose that a : R + R + is monotonically decreasing. Then
n = 2 a ( n log n )  converges  e + a ( x log x ) d x  converges. 
Now due to the monotonicity of a, we have
e + a ( x log x ) d x = e + a ( t log t log ( t log t ) ) ( log t + 1 ) d t e + a ( t ) log t d t .
and
e + a ( x log x ) d x = 2 1.98 + a ( 2 t log t log ( 2 t log t ) ) ( log t + 1 ) d t 2 1.98 + a ( t ) ( log t + 1 ) d t 6 1.98 + a ( t ) log t d t .
Thus, if a log is also monotonically decreasing, then
n = 2 a ( n log n )  converges  n = 2 a ( n ) log n  converges. 

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