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 \(f(x)\).
Try Part II.
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