Manipulated Taylor Series

Calculus Level 1

Determine the Taylor series centered about $x = 0$ for the function

$f(x) = 2x^3\sin 4x^5.$

\sum_{n = 0}^{\infty}(-1)^n \frac{2^{4n + 3} x^{10n + 5}}{(2n + 1)!}

\sum_{n = 0}^{\infty}(-1)^n \frac{4^{2n + 2} x^{10n + 5}}{(2n + 1)!}

\sum_{n = 0}^{\infty}(-1)^n \frac{2^{4n + 3} x^{10n + 8}}{(2n + 1)!}

\sum_{n = 0}^{\infty}(-1)^n \frac{4^{2n + 2} x^{10n + 8}}{(2n + 1)!}