# To test the convergence $\sum{\frac{1}{(\ln {n})^{\ln{n}}}}$ [duplicate]

How to test the convergence of following series $$\sum{\frac{1}{(\ln {n})^{\ln{n}}}}$$

I have tried Cauchy condensation test and gives me nothing

• See this. – David Mitra Jan 22 '17 at 22:37
• @DavidMitra Perfect find :D – Simply Beautiful Art Jan 22 '17 at 22:38
• @SimplyBeautifulArt It was the second "related" question :) – David Mitra Jan 22 '17 at 22:39
• @DavidMitra >.> Well at least I didn't use a totally copied answer from that place. – Simply Beautiful Art Jan 22 '17 at 22:40

Method 1: Claim: $(\ln n)^{\ln n} > n^2$ for $n>e^{e^2}.$ The claim implies

$$\frac{1}{(\ln n)^{\ln n}} < \frac{1}{n^2}$$

for these $n.$ Since $\sum 1/n^2$ converges, so does $\sum 1/(\ln n)^{\ln n}.$

To prove the claim, apply $\ln$ to both sides. I'll leave this to the reader for now.

Method 2: Cauchy condensation: Consider the series

$$\tag 1\sum \frac{2^n}{(\ln 2^n)^{\ln 2^n}} = \sum \frac{2^n}{(n\ln 2)^{n\ln 2} }= \sum \frac{2^n}{((n\ln 2)^{\ln 2})^n}.$$

Now $(n\ln 2)^{\ln 2}\to \infty,$ so eventually $(n\ln 2)^{\ln 2} \ge 4.$ This shows the terms in $(1)$ are eventually bounded above by $2^n/4^n = 1/2^n.$ This proves convergence of the original series.

Let's try that Cauchy condensation test again with a ratio test:

$$\frac1{(\ln n)^{\ln n}}\implies\frac{2^n}{(\ln2^n)^{\ln2^n}}=\frac{2^n}{(cn)^{cn}}\implies\frac{(2/c(n+1))^{c(n+1)}}{(2/cn)^{cn}}\to0$$

where $c=\ln2$.

Thus, it converges.

• $\frac{2^n}{(\ln2^n)^{\ln2^n}}<\frac{2^n}{(\ln e^n)^{\ln e^n}}$ Are you sure it's ok, since $\ln{2^n}<\ln{e^n}$ ? – UfmdFkiF Jan 22 '17 at 22:42
• @TheMeff Oh, whoops, that went wrong. But either way... – Simply Beautiful Art Jan 22 '17 at 22:43
• Usually $\implies$ means "implies". – zhw. Jan 22 '17 at 23:10