Mathematics

Problem

Let

h(x)=\displaystyle\int_{0}^{x} t^{2}e^{t^{3}}\,dt

. a. Expand

e^{t^{3}}

in a Maclaurin series and write its general term. b. Form the expanded integrand

t^{2}e^{t^{3}}

and write its general term. c. Integrate term by term to obtain a Maclaurin series for

h(x)

. Express the general term of the resulting series in closed form. d. Identify the coefficient of

x^{3k+3}

for a non-negative integer

k

and write it explicitly in terms of

k

. e. Determine the radius of convergence of the Maclaurin series for

h(x)

and justify your answer.