Mathematics

Problem

Define

g(x)=e^{x}\,\ln(1+x)

. a. Write the first eight non-zero terms of the Maclaurin series for

e^{x}

and the first eight non-zero terms for

\ln(1+x)

. b. Using the two series from part (a), determine—by collecting terms—the exact coefficient of

x^{7}

in the Maclaurin series for

g(x)

. Clearly indicate each pair of terms whose product contributes to the

x^{7}

term. c. Verify that the coefficient you found in part (b) agrees with the coefficient obtained by differentiating your product series and evaluating at

x=0

. Provide a brief explanation of why the two approaches must yield the same value.

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