Mathematics

Problem

Recall the Maclaurin series of

\ln(1+x)

and

\sin x

. Consider the function

Q(x)=\ln(1+x)-\sin x

. (a) Determine the first four nonzero terms of the Maclaurin series of

Q(x)

and state the corresponding coefficients. (b) Using your series from part (a), evaluate

\displaystyle \lim_{x\to0}\frac{\ln(1+x)-\sin x}{x^{3}}

. Provide complete justification connecting the limit to a specific series coefficient.