Q 19 - 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.

$\frac{1}{1-x},x_0=0$

Step-by-Step Solution

Answer

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

Step 1: Given information

The function is $f\left(x\right)=\frac{1}{1-x}$

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 generic form Maclurin series can be written.

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

Step 3: Make a table of the Maclurin series for the function f ( x ) = 1 1 - x   at x = 0  

$n$	$f^n\left(x\right)$	$f^n\left(0\right)$	$\frac{f^n\left(0\right)}{n!}$
$0$	$\frac{1}{1-x}$	$1$	$1$
$1$	$\frac{1}{{(1-x)}^2}$	$1$	$1$
$2$	$\frac{2}{(1-x{)}^3}$	$2$	$1$
$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$	$$\begin{gathered} . \\ . \\ . \end{gathered}$$
$k$	$\frac{k!}{(1-x{)}^{k+1}}$	$k !$	$1$

Step 4: Find the Maclaurin series for the function f ( x ) = 1 1 - x   at x = 0  

 The Maclaurin series for the function $f\left(x\right)=\frac{1}{1-x}$at $x=0$is:

$1+x+x^2+x^3+\cdots +x^k+\cdots$

Or we can write as:

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