Question

Find the indicated Maclaurin or Taylor series for the given function about the indicated point, and find the radius of convergence for the series.

$e^x,x_0=0$

Short answer

The Maclurin series for the function is $f\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{x}^k$

Step-by-step solution

Given information

The function is$f\left(x\right)=e^x$

 Find the general of the Maclurin series of the function 

The Maclurin series at $x_0=0$for any function $f$with a derivative of all orders is given by

$f\left(x\right)=f\left(0\right)+f^{\text{'}}\left(0\right)x+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}{x}^2+\frac{f^{\text{'}\text{'}\text{'}}\left(0\right)}{3!}{x}^3+\frac{f^{\text{'}\text{'}\text{'}\text{'}}\left(0\right)}{4!}{x}^4+\dots$

The function's general Maclurin series is$f\left(x\right)=\sum _{n=0}^{\infty }\frac{f^n\left(0\right)}{n!}{x}^n$

 Make a table of the Maclurin series for the function f(x)=ex 

$n$	$f^n\left(x\right)$	$f^n\left(0\right)$	$\frac{f^n\left(0\right)}{n!}$
$0$	$e^x$	$1$	$1$
$1$	$e^x$	$1$	$1$
$2$	$e^x$	$1$	$\frac{1}{2!}$
$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$
$k$	$e^x$	$1$	$\frac{1}{k!}$

 Find the Maclurin series for the function f(x)=ex 

The Maclurin series for the function $f\left(x\right)=e^x$is:

$1+x+\frac{1}{2!}{x}^2+\frac{1}{3!}{x}^3+\frac{1}{4!}{x}^4+\dots$

Or, we can write as:

$f\left(x\right)=\sum _{k=0}^{\infty }\frac{1}{k!}{x}^k$