convergent or divergent

[spring]

does
\(\displaystyle \sum_{n=2}^\infty \dfrac{1}{(lnn)^2}\)
converge or diverge? how?
also, what is the problem with my latex code?

 

[Prove It]

does
\(\displaystyle \sum_{n=2}^\infty \dfrac{1}{(lnn)^2}[\math]
converge or diverge? how?
also, what is the problem with my latex code?\)

\(\displaystyle

I believe that this is convergent, due to it being an over-harmonic series...

Also, your code should say \frac, not \dfrac\)

 

[Failure]

does
\(\displaystyle \sum_{n=2}^\infty \dfrac{1}{(\ln n)^2}[\math]
converge or diverge? how?
also, what is the problem with my latex code?\)

\(\displaystyle

This series diverges by the Cauchy condensation test - Wikipedia, the free encyclopedia

\(\displaystyle \sum_n 2^n\frac{1}{\ln^2 2^n}=\sum_n \frac{2^n}{n^2\ln 2}=+\infty\)

As regards your LaTeX code: your LaTeX code was ok, but the closing math tag had a backward instead a forward slash. You really do need to use a forward slash in the closing math tag, like this: \(\displaystyle \ldots [{\color{red}/}\text{math}]\)\)

 

[spring]

pöf!
backward slash of course! thank you for your help dear Failure.
it is a fine way to use cauchy condensation test for this, but i think its divergence is also proven by limit comparison test with \(\displaystyle \sum_n \frac{1}{n}\) i have just discovered it!


by the way, thank you for your believing dear Prove It %)