Integrate & differentiate power series

Practice integrating and differentiating power series, or analyzing the integrals/derivatives of power series.

Problem

f(x)=n=0(1)n+12x2n+1(2n+1)!\displaystyle f(x)=\sum_{n=0}^{\infty }{{{\left( -1 \right)}^{n+1}}\frac{2{{x}^{2n+1}}}{\left( 2n+1 \right)!}}
f, start superscript, prime, prime, prime, end superscript, left parenthesis, 0, right parenthesis, equals
  • 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