Let and , each of which converges for . Which statement about the Maclaurin series of…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

f(x)=\displaystyle\sum_{n=0}^{\infty}a_n x^{n}

and

g(x)=\displaystyle\sum_{n=0}^{\infty}b_n x^{n}

, each of which converges for

|x|<2

. Which statement about the Maclaurin series of

h(x)=f(x)+g(x)

and

k(x)=f(x)g(x)

is true? (a) The series for

h(x)

converges for

|x|<2

, but the series for

k(x)

converges only for

|x|<1

. (b) Both series for

h(x)

and

k(x)

converge for all real

x

.},{ (c) Both series for

h(x)

and

k(x)

converge for

|x|<2

.},{ (d) The series for

h(x)

converges for

|x|<1

, while the series for

k(x)

converges for

|x|<2

.}],