Mathematics

Problem

The Maclaurin series for

\cos x

\cos x = 1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots

which converges for all real numbers

x

. (a) Using the first three non-zero terms of the series, calculate an approximation to

\cos(0.3)

. Express your answer to at least six decimal places. (b) The remainder after the sixth–degree term satisfies

|R_3(x)|\le\dfrac{|x|^{8}}{8!}

for any real

x

. Justify that the approximation obtained in part (a) is accurate to six decimal places.