009C Sample Final 2, Problem 7

(a) Consider the function  $f(x)={\bigg (}1-{\frac {1}{2}}x{\bigg )}^{-2}.$  Find the first three terms of its Binomial Series.

(b) Find its radius of convergence.

Foundations:
1. The Taylor polynomial of   $f(x)$   at   $a$   is
$\sum _{n=0}^{\infty }c_{n}(x-a)^{n}$ where $c_{n}={\frac {f^{(n)}(a)}{n!}}.$
2. Ratio Test
Let  $\sum a_{n}$  be a series and  $L=\lim _{n\rightarrow \infty }{\bigg |}{\frac {a_{n+1}}{a_{n}}}{\bigg |}.$
Then,
If  $L<1,$  the series is absolutely convergent.
If  $L>1,$  the series is divergent.
If  $L=1,$  the test is inconclusive.

Solution:

(a)

Step 1:
We begin by finding the coefficients of the Maclaurin series for  $f(x)={\frac {1}{(1-{\frac {1}{2}}x)^{2}}}.$
We make a table to find the coefficients of the Maclaurin series.
$n$ $f^{(n)}(x)$ $f^{(n)}(0)$ ${\frac {f^{(n)}(0)}{n!}}$ $0$ ${\frac {1}{(1-{\frac {1}{2}}x)^{2}}}$ $1$ $1$ $1$ ${\frac {1}{(1-{\frac {1}{2}}x)^{3}}}$ $1$ $1$ $2$ ${\frac {\frac {3}{2}}{(1-{\frac {1}{2}}x)^{4}}}$ ${\frac {3}{2}}$ ${\frac {3}{4}}$

Step 2:
So, the first three terms of the Binomial Series is
$1+x+{\frac {3}{4}}x^{2}.$

(b)

Step 1:
By taking the derivative of the known series
${\frac {1}{1-x}}\,=\,1+x+x^{2}+\cdots ,$
we find that the Maclaurin series of  ${\frac {1}{(1-x)^{2}}}$  is
$\sum _{n=0}^{\infty }(n+1)x^{n}.$
Letting $x/2$   play the role of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,} the Maclaurin series of  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{(1-\frac{1}{2}x)^2}}   is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=0}^\infty (n+1)\bigg(\frac{1}{2}x\bigg)^n=\sum_{n=0}^\infty \frac{(n+1)x^n}{2^n}.}

Step 2:
Now, we use the Ratio Test to determine the radius of convergence of this power series.
We have
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{\lim_{n\rightarrow \infty} \bigg|\frac{a_{n+1}}{a_n}\bigg|} & = & \displaystyle{\lim_{n\rightarrow \infty} \bigg| \frac{(n+2)x^{n+1}}{2^{n+1}} \frac{2^n}{(n+1)x^n}\bigg|}\\ &&\\ & = & \displaystyle{\lim_{n\rightarrow \infty} \frac{|x|}{2} \frac{n+2}{n+1}}\\ &&\\ & = & \displaystyle{\frac{|x|}{2}\lim_{n\rightarrow \infty}\frac{n+2}{n+1}}\\ &&\\ & = & \displaystyle{\frac{|x|}{2}.} \end{array}}
Now, the Ratio Test says this series converges if  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{|x|}{2}<1.}   So,  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|<2.}
Hence, the radius of convergence is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=2.}

Final Answer:
(a)    Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+x+\frac{3}{4}x^2}
(b)    The radius of convergence is  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R=2.}

Return to Sample Exam