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 $f^{(n)} (x)$ .

**Details and Assumptions:**

- $f^{(n)}(x)$ denotes the $n^{th}$ derivative of $f(x)$.