Maclaurin series for sin x, cos x, and eˣ

Maclaurin series for sin x, cos x, and eˣ practice problems.

Problem

Find the exact value of the sum of the infinite series given below.
22π22!+2π44!...+2(1)nπ2n(2n)!+...\qquad\displaystyle2 -\frac{{{2\pi }^{2}}}{2!}+\frac{{2{\pi }^{4}}}{4!}-...+{{2\left( -1 \right)}^{n}}\frac{{{\pi }^{2n}}}{\left( 2n \right)!}+...
  • Your answer should be
  • an integer, like 6
  • an exact decimal, like 0, point, 75
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4