Given $f(x)$ with base point $c,$ the Taylor series at base point $c$ is the power series

${\displaystyle \underset{k=0}{\overset{}{}}}{a}_k(x-c{)}^k$

$($a$)$ State the formula for $a_k.$

$($b$)$ State the MaCl aurin series of the following functions in the summation form, and their radii of convergence:

$e^x,cos(x),sin(x)$

$($c$)$ State the Taylor series of $f(x)=sin(x)$ at $c=\frac{}{4}$ up to degree $5.$Given $f(x)$ with base point $c,$ the Taylor series at base point $c$ is the power series

${\displaystyle \underset{k=0}{\overset{}{}}}{a}_k(x-c{)}^k$

$($a$)$ State the formula for $a_k.$

$($b$)$ State the MaCl aurin series of the following functions in the summation form, and their radii of convergence:

$e^x,cos(x),sin(x)$

$($c$)$ State the Taylor series of $f(x)=sin(x)$ at $c=\frac{}{4}$ up to degree $5.$Given $f(x)$ with base point $c,$ the Taylor series at base point $c$ is the power series

${\displaystyle \underset{k=0}{\overset{}{}}}{a}_k(x-c{)}^k$

$($a$)$ State the formula for $a_k.$

$($b$)$ State the MaCl aurin series of the following functions in the summation form, and their radii of convergence:

$e^x,cos(x),sin(x)$

$($c$)$ State the Taylor series of $f(x)=sin(x)$ at $c=\frac{}{4}$ up to degree $5.$