Derivative Magic I

Calculus Level 5

Define $f(x)=\displaystyle\sum_{k=1}^{1001} \dfrac{x^{k+1}}{k}$ . If $f^{(1001)} (100)$ can be evaluated as $a\times b!$ where $b$ is maximum, find $a+b$ .

Bonus: If $f(x)=\displaystyle\sum_{k=1}^{n} \dfrac{x^{k+1}}{k}$ , find $f^{(n)} (x)$ .