Prove a divergent sequence exists

Question

Show that a positive sequence {$a_i$} exists (or does not exist) such that both of the following are true:

1. $\lim_{i\to\infty}a_i=+\infty$
2. $\lim_{n\to\infty}\;\sum_{i=1}^{n}\frac{2}{a_i}-\sqrt{n\ln(n)}=+\infty$

My thoughts: I think {$a_i$} does exist: if we "ignore" the first condition and assume $a_i$ is a constant, then $\lim_{n\to\infty}\;\sum_{i=1}^{n}\frac{2}{C}-\sqrt{n\ln(n)}=2n/C-\sqrt{n\ln(n)}=+\infty$. I figured if we can make $a_i$ increase really slow, then this might work but I'm not sure. Certainly, we can try $a_i=\ln(i)$ but I don't know how to evaluate the partial sum of $1/\ln(i)$. Is there any place I can look up for lower bound of partial sums by the way? Any help will be appreciated.

Answer 1

What happens if we fix $0 < \beta < 1$ and define $$ a_k = k^\beta \; ? $$ How does your question part 2. change as we try different values of $\beta?$

Easiest method for estimating a sum: