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I was trying to determine if the following infinite sum (from 1 to infinity) diverges or convergence, but I am completely stuck. What test would be most appropriate? I know that $\ln k <k$ and that $\ln \ln k<k$ for $k<10$. Any help would be appriciated!

$\sum \frac{1}{{(\ln k)} ^{\ln \ln k}}$

William
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  • First of all, you probably want the sum to start at $k=2$, otherwise $\log\log k$ is not well defined. A quick Python script calculating partial sums to N tells me it is probably diverging: $N=10$ gives 6,54, $N=100$ is 22,42, $N=1000$ is 57,93, $N=10000$ is 157,01, and $N=100000$ is 487,47. – SomeCallMeTim Mar 07 '22 at 10:05

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The Cauchy condensation test says your convergence is equivalent to the convergence of $$\sum e^nn^{-\ln n}=\sum e^{n-\ln^2n}.$$ Clearly these terms do not even decay to zero, let alone converge as a series.

Funktorality
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