Taylor Series

Determine the Taylor series for the function

$f(x)=\int _0 ^1 \dfrac{\sin x}{x}dx ~\text{ centered at }~ x=0.$

\sum_{n=0}^{\infty} \dfrac{(-1)^n }{2n (2n+1)! }

\sum_{n=0}^{\infty} \dfrac{(-1)^n }{(2n+2)!}

\sum_{n=0}^{\infty} \dfrac{(-1)^n }{(2n+1)! (2n+1)}

\sum_{n=0}^{\infty} \dfrac{(-1)^n }{(2n)!}

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

$f(x)=\sin x\cos x ~\text{ centered at }~ x=0.$

\sum_{n = 0}^{\infty}\frac{x^{2n-1}}{(2n-1)}

\dfrac{1}{2} \sum_{n=0}^{\infty} \dfrac{(-1)^n (2x)^{2n+1}}{(2n+1)!}

2 \sum_{n=0}^{\infty} \dfrac{(-1)^n (2x)^{2n+1}}{(2n+1)!}

\sum_{n=0}^{\infty} \dfrac{(-1)^n (2x)^{2n+1}}{(2n+1)!}

Evaluate

$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n \cdot 2^n }.$

If $f(x)=e^{x^2}$ , what is $f^{(2016)}(0)$ ?

Taylor Series - Problem Solving