Question

# Examine whether the series ∞∑ 1/(logn)^logn∑ is convergent.n=2

Logarithms

Examine whether the series $$\sum_1^∞=(\log n)^{\log n}$$ is convergent.

$$\displaystyle{n}={2}$$

2021-06-24

Notice that, for all $$\displaystyle{n}\Rightarrow{2}$$
$$\displaystyle{\left({\log{{n}}}\right)}^{{\log{{n}}}}={e}^{{\log{{\left({\left({\log{{n}}}\right)}^{{\log^{n}}}\right)}}}}={\left({e}^{{\log{{n}}}}\right)}^{{\log{{\left({\log{{n}}}\right)}}}}={n}^{{\log{{\left({\log{{n}}}\right)}}}}$$
Also note that $$\displaystyle{\log{{\left({\log{{n}}}\right)}}}{>}{2}$$ for all $$\displaystyle{n}{>}{e}^{{e^{2}}}$$.

Choose n0 such that $$\displaystyle{n}{0}{>}{e}^{{e^{2}}}$$.

Then for all $$\displaystyle{n}\Rightarrow{n}{0}$$ we have $$\displaystyle\frac{{1}}{{{\left({\log{{n}}}\right)}^{{\log{{n}}}}}}=\frac{{1}}{{{n}^{{\log}}}}{\left({\log{{n}}}\right)}\le\frac{{1}}{{n}^{{2}}}$$
The given series is convergent by comparison test.