Derivative Magic I

Calculus Level 5

Define f(x)=k=11001xk+1kf(x)=\displaystyle\sum_{k=1}^{1001} \dfrac{x^{k+1}}{k} . If f(1001)(100)f^{(1001)} (100) can be evaluated as a×b!a\times b! where bb is maximum, find a+ba+b .

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

Details and Assumptions:

  • f(n)(x)f^{(n)}(x) denotes the nthn^{th} derivative of f(x)f(x).
Try Part II.
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