# 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 \(f^{(n)} (x)\) .

**Details and Assumptions:**

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