Compute the limit $$\lim _{n \rightarrow \infty} \left( \frac{1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}}{\ln n} \right) ^{\ln n}$$
I know how to do $$\lim _{n \rightarrow \infty} \frac{H_N}{\ln (n)} = 1$$ via L'Hopital's rule, but I'm not quite sure whether or not something similar is applicable here. I'm particularly concerned about the growth rates of the numerator and denominator in this case, but I haven't been able to make any meaningful progress. I would greatly appreciate assistance on this problem. Thank you.
essentially $$ \left( 1 + \frac{\gamma}{\log n} \right)^{\log n} $$ which gives $$ e^\gamma $$