I have next problem: $$ e^{H_n} = ... + O(\frac1n)$$
$H_n = \sum_{k=1}^n \frac1k $
I have got the following result:
From ($ H_n > \ln n + \gamma $) I go to $e^{H_n} \geq n e^{\gamma}$
Therefore $e^{H_n} = n e^{\gamma} + ... + O(\frac1n)$
What is the next step to get remaining elements of the sum?