Root Test: Convergence of Series

[azatkgz]

Homework Statement

Determine whether the series converges or diverges.

$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln (\ln n)}}=\sum_{n=2}^{\infty}\frac{1}{e^{\ln (\ln n)\ln (\ln n)}}$$

The Attempt at a Solution

for $$u=\ln (\ln n)$$

$$\sum_{u=\ln (\ln 3)}^{\infty}\frac{1}{e^{u^2}}$$

from Root Test

$$\lim_{u\rightarrow\infty}\sqrt<u>{\frac{1}{e^{u^2}}}=\lim_{u\rightarrow\infty}\frac{1}{e^u}=0<1</u>$$
series converges

 

[quasar987]

That u substitution does not give back the original series. ln(ln(3))+1 does noe equal ln(ln(4)).

Try simply the comparison criterion. Hint: (logx)²/x -->0, so (logx)²<x for x large enough.

 

[azatkgz]

Try simply the comparison criterion. Hint: (logx)²/x -->0, so (logx)²<x for x large enough.

$$\ln^2x<x$$

for x=lnn

$$\ln^2(\ln n)<\ln n$$

$$e^{\ln^2(\ln n)}<e^{\ln n}=n$$

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

diverges

 

[quasar987]

good!