Consider the series

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

Problem

Consider the series

S(x)=\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}

. Which of the following statements best explains why

S(x)=\cos x

for every real number

x

? (a) Because

S''(x)=-S(x)

for all

x

, any function satisfying this differential equation must be

\cos x

. (b) The coefficients of

S(x)

match the derivatives of

\cos x

x=0

divided by factorials, and the series converges for all

x

, so it is the Taylor series of

\cos x

. (c) The ratio test shows

S(x)

converges for all

x

, and any absolutely convergent even series equals

\cos x

. (d) Since the terms of

S(x)

alternate in sign and decrease in magnitude, the partial sums must oscillate between

-1

and

1

, matching the range of

\cos x