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 $f^{(2016)}(0)$ ?