Calculus
Taylor Series
Determine the Taylor series for the function
\[f(x)=\int _0 ^1 \dfrac{\sin x}{x}dx ~\text{ centered at }~ x=0.\]
Evaluate \[\frac{\pi^2}{2!}-\frac{\pi^4}{4!}+\frac{\pi^6}{6!}-\frac{\pi^8}{8!}+\cdots.\]
Determine the Taylor series for the function
\[f(x)=\sin x\cos x ~\text{ centered at }~ x=0.\]
Evaluate
\[ \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \cdot 2^n }. \]
If \(f(x)=e^{x^2}\), what is \(f^{(2016)}(0)\) ?