For what values for m does ∑ k = 2 ∞ 1 ( ln ⁡ k ) m converge?

Question

For what values for m does      ∑          k      =      2              ∞            1          (      ln      ⁡              k                    )        m            converge?

Hayley Mccarthy · Accepted Answer

For your first seriesLet   α  ,  β  ∈      R        ∑          k      =      2        ∞        1                  k        α            ln      ⁡      (      k              )        β            is convergent iif   (  α  &amp;gt;  1  ) or   (  α  =  1  ,     β  &amp;gt;  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              &amp;gt;  1which yields      1          ln      ⁡      (      k              )        β              &amp;gt;      1          k      The right-hand side behind divergent, your series is divergent. These series are called Bertrand series.

For what values for m does <munderover> <mo movablelimits="false">∑ <mr

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