The problem is stated as following:
Determine $\lim_{n\rightarrow \infty} \frac{1}{2\ln(2)} + \frac{1}{3\ln(3)} + \dots + \frac{1}{n\ln(n)} - \ln(\ln(n))$
My attempt:
First of all, I try to simplify the expression to:
$\displaystyle L = \lim_{n\rightarrow \infty} \left [\sum_{k=2}^{n}\frac{1}{k\ln(k)} \right ]- \ln(\ln(n))$
We notice that $\displaystyle \sum_{k=2}^{n}\frac{1}{k\ln(k)}$ can be approximated to $\displaystyle\lim_{n\rightarrow \infty}\int_ {2}^{n}\frac{dx}{x\ln(x)} = \lim_{n\rightarrow \infty} \ln(\ln(n))-\ln(\ln(2))$
Plugging that into our first expression we get:
$\displaystyle\lim_{n\rightarrow \infty} -\ln(\ln(2)) = -\ln(\ln(2))$. Hence:
$$\boxed{L = -\ln(\ln(2))}$$
Comments:
I don't know whether it's okay to approximate the series with this limit without stating whether the integral is greater than or equal to the first expression. If I had to do that, then I'd also have to show that the expression is greater than 0, in order to use the limit comparision test.
I'd be glad if you could add any comments on what (if any) steps in my solution went wrong, and why in that case.
Thanks!
