1
Montrer qu’il existe \(C\in\R\) tel que \(\sum_{k=1}^n \frac{\ln k}k = \frac12\ln^2(n) + C + o(1)\).
2
Prouver : \(\frac{\ln2}2 - \int_{t=1}^3\frac{\ln t}t\,d t \le C \le \frac{\ln2}2 + \frac{\ln3}3 - \int_{t=1}^3\frac{\ln t}t\,d t\).
3
Prouver : \(\sum_{k=1}^n \frac{\ln k}k = \frac12\ln^2(n) + C + \frac{\ln n}{2n} + o\Bigl(\frac{\ln n}{n}\Bigr)\).