The task is to find asymptotic behavior of sum: $\sum\limits_{k=2}^{m}\frac{1}{\ln(k!)}$ when $m\to\infty$.
Any help with solving this one?
The task is to find asymptotic behavior of sum: $\sum\limits_{k=2}^{m}\frac{1}{\ln(k!)}$ when $m\to\infty$.
Any help with solving this one?
Using Stirling's approximation: $\ln(n!)\sim n\ln(n)+O(n)$
Next we approximate sum with integral: $\sum\limits_{k=2}^{m}\frac{1}{k\ln(k)}\sim\int_{2}^{m}\frac{dx}{x\ln(x)}=\ln\ln(m)-\ln\ln(2)$
Found asymptotic behavior — $\ln \ln(n)$.