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n-th derivative of lnx/x

Source: ISI(BM) 2009 #6

May 6, 2012
calculusderivativefunctionlogarithmsintegrationcalculus computations

Problem Statement

Let f(x)f(x) be a function satisfying xf(x)=ln⁡x        for  x>0xf(x)=\ln x \ \ \ \ \ \ \ \ \text{for} \ \ x>0 Show that f(n)(1)=(−1)n+1n!(1+12+⋯+1n)f^{(n)}(1)=(-1)^{n+1}n!\left(1+\frac{1}{2}+\cdots+\frac{1}{n}\right) where f(n)(x)f^{(n)}(x) denotes the nn-th derivative evaluated at xx.
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