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Michelle Mendoza 2022-07-03 Answered
For what values for m does
k = 2 1 ( ln k ) m
converge?
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Answers (1)

Hayley Mccarthy
Answered 2022-07-04 Author has 19 answers
For your first series
Let α , β R
k = 2 1 k α ln ( k ) β
is convergent iif ( α > 1 ) or ( α = 1 ,   β > 1 )
So in your case it is not convergent. As
k ln ( k ) β
we can find k 0 such that for all k k 0
k ln ( k ) β > 1
which yields
1 ln ( k ) β > 1 k
The right-hand side behind divergent, your series is divergent. These series are called Bertrand series.
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