Consider the function defined for by

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Consider the function

g

defined for

x>0

g(x)=\dfrac{\ln x}{x}

. The function is to be analyzed near

x=1

. (a) Starting from a known Taylor series, write the first four nonzero terms of the Taylor series of

\ln x

about

x=1

. Express each term in powers of

(x-1)

. (b) Use your result from part (a) to obtain the first four nonzero terms of the Taylor series of

g

about

x=1

. Simplify each coefficient. (c) From your series in part (b), determine the exact values of

g''(1)

and

g^{(4)}(1)

. Clearly indicate how each derivative is obtained from the series.