Does the series $\sum _{n=1} ^{\infty} P\big(\xi_n > \sqrt{2 \ln n + 2 \ln \ln n} \big)$ converge?

Question

Let $\{\xi_n\}_{n\ge1}$ be a series of independent normally distributed random variables with $\mu = 0$ and $\sigma = 1$.

Does the series $\sum _{n=1} ^{\infty} P\big(\xi_n > \sqrt{2 \ln n + 2 \ln \ln n} \big)$ converge?

Attempt Applied Chebyshev's inequality and got $$P\big(\xi_n > \sqrt{2 \ln n + 2 \ln \ln n} \big) \le \frac{1}{2}\frac{1}{2 \ln n + 2 \ln \ln n} $$ so the series in question converges if $$\sum _{n=1} ^{\infty} \frac{1}{2}\frac{1}{2 \ln n + 2 \ln \ln n}$$ does. But then I got stuck trying to prove this latter statement.

Answer 1

Hint: The series you mention, as pointed out in a comment, diverges. But you probably gave up a lot using Chebyshev. You can get a better estimate using L'Hopital on

$$ \lim_{x\to \infty}\frac{\int_x^\infty e^{-t^2/2}\,dt}{e^{-x^2/2}/(2x)}.$$