Define a function $sinc(x)($pronounced "sink of $x")$ by:

$sinc(x)=\{\begin{array}{ccc}\frac{sin(x)}{x}& if& x0\\ 1& if& x=0\end{array}$

$($This function is used frequently in electrical engineering and signal processing.$)$

Use this list of Basic Taylor Series to find the Taylor Series for $f(x)=sinc(x)$ based at $0.$ Give your answer using summation notation and give the largest open interval on which the series converges. $($If you need to enter $,$ use the $$ button in CalcPad or type "infinity" in all lower$-$case.$)$

The Taylor series for $sinc(x)$ is: ${\displaystyle \underset{k=0}{\overset{}{}}}$

The Taylor series converges to $sinc(x)$ for $|x|<mtext></mtext><mi></mi>$

Use the Taylor series you just found for $sinc(x)$ to find the Taylor series for $f(x)={\displaystyle \underset{0}{\overset{x}{}}}sinc(t)dt$ based at $0.$

Give your answer using summation notation and give the largest open interval on which the series converges.

The Taylor series for $f(x)={\displaystyle \underset{0}{\overset{x}{}}}sinc(t)dt$ is: $$

$$

The Taylor series converges to $f(x)$ for $|x|$