The natural logarithm function satisfies for

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

Problem

The natural logarithm function satisfies

f(x)=\ln(1+x)

for

-1<x\le 1

. Consider the related function

h(x)=\ln(1-4x)

, which is also centered at

x=0

. (a) Express

h(x)

explicitly in terms of

f

. (b) The Maclaurin series of

f

f(x)=\sum_{n=1}^{\infty}c_nx^n

. Describe precisely how each coefficient in the Maclaurin series of

h

can be written in terms of

c_n

. State a general formula for the coefficient of

x^{n}

in the series for

h

. (c) Determine the interval of convergence of the Maclaurin series for

h(x)

and justify your conclusion using an appropriate inequality.