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.
$\qquad\displaystyle2 -\frac{{{2\pi }^{2}}}{2!}+\frac{{2{\pi }^{4}}}{4!}-...+{{2\left( -1 \right)}^{n}}\frac{{{\pi }^{2n}}}{\left( 2n \right)!}+...$