Q 17 - Calculus

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. 

$e^x,x_0=0$

Step-by-Step Solution

Answer

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

Step 1: Given information

The function is $f\left(x\right)=e^x$

Step 2: 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 $f\left(x\right)=\sum _{n=0}^{\infty }\frac{f^n\left(0\right)}{n!}{x}^n$

Step 3: Make a table of the Maclurin series for the function f ( x ) = e x  

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

Step 4: Find the Maclurin series for the function f ( x ) = e x  

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:

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