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The applications of Taylor series in this section are intended to highlight their importance. In general, Taylor series are useful because they allow us to represent known functions using polynomials, thus providing us a tool for approximating function values and estimating complicated integrals. In addition, they allow us to define new functions as power series, thus providing us with a powerful tool for solving differential equations.
In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial.
${\left(1-x\right)}^{1\text{/}3}$
${\left(1+{x}^{2}\right)}^{\mathrm{-1}\text{/}3}$
${\left(1+{x}^{2}\right)}^{\mathrm{-1}\text{/}3}={\displaystyle \sum _{n=0}^{\infty}\left(\begin{array}{c}-\frac{1}{3}\hfill \\ \hfill n\hfill \end{array}\right){x}^{2n}}$
${\left(1-x\right)}^{1.01}$
${\left(1-2x\right)}^{2\text{/}3}$
${\left(1-2x\right)}^{2\text{/}3}={\displaystyle \sum _{n=0}^{\infty}{\left(\mathrm{-1}\right)}^{n}{2}^{n}}\left(\begin{array}{c}\frac{2}{3}\hfill \\ n\hfill \end{array}\right){x}^{n}$
In the following exercises, use the substitution ${\left(b+x\right)}^{r}={\left(b+a\right)}^{r}{\left(1+\frac{x-a}{b+a}\right)}^{r}$ in the binomial expansion to find the Taylor series of each function with the given center.
$\sqrt{x+2}$ at $a=0$
$\sqrt{{x}^{2}+2}$ at $a=0$
$\sqrt{2+{x}^{2}}={\displaystyle \sum _{n=0}^{\infty}{2}^{\left(1\text{/}2\right)-n}}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){x}^{2n};\left(\left|{x}^{2}\right|<2\right)$
$\sqrt{x+2}$ at $a=1$
$\sqrt{2x-{x}^{2}}$ at $a=1$ ( Hint: $2x-{x}^{2}=1-{\left(x-1\right)}^{2})$
$\sqrt{2x-{x}^{2}}=\sqrt{1-{\left(x-1\right)}^{2}}$ so $\sqrt{2x-{x}^{2}}={\displaystyle \sum _{n=0}^{\infty}{\left(\mathrm{-1}\right)}^{n}}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-1\right)}^{2n}$
${\left(x-8\right)}^{1\text{/}3}$ at $a=9$
$\sqrt{x}$ at $a=4$
$\sqrt{x}=2\sqrt{1+\frac{x-4}{4}}$ so $\sqrt{x}={\displaystyle \sum _{n=0}^{\infty}{2}^{1-2n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-4\right)}^{n}}$
${x}^{1\text{/}3}$ at $a=27$
$\sqrt{x}$ at $x=9$
$\sqrt{x}={\displaystyle \sum _{n=0}^{\infty}{3}^{1-3n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-9\right)}^{n}}$
In the following exercises, use the binomial theorem to estimate each number, computing enough terms to obtain an estimate accurate to an error of at most $1\text{/}1000.$
[T] ${\left(15\right)}^{1\text{/}4}$ using ${\left(16-x\right)}^{1\text{/}4}$
[T] ${\left(1001\right)}^{1\text{/}3}$ using ${\left(1000+x\right)}^{1\text{/}3}$
$10{\left(1+\frac{x}{1000}\right)}^{1\text{/}3}={\displaystyle \sum _{n=0}^{\infty}{10}^{1-3n}}\left(\begin{array}{c}\frac{1}{3}\hfill \\ n\hfill \end{array}\right){x}^{n}.$ Using, for example, a fourth-degree estimate at $x=1$ gives $\begin{array}{cc}\hfill {\left(1001\right)}^{1\text{/}3}& \approx 10\left(1+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 1\hfill \end{array}\right){10}^{\mathrm{-3}}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 2\hfill \end{array}\right){10}^{\mathrm{-6}}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 3\hfill \end{array}\right){10}^{\mathrm{-9}}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 4\hfill \end{array}\right){10}^{\mathrm{-12}}\right)\hfill \\ & =10\left(1+\frac{1}{{3.10}^{3}}-\frac{1}{{9.10}^{6}}+\frac{5}{{81.10}^{9}}-\frac{10}{{243.10}^{12}}\right)=\mathrm{10.00333222...}\hfill \end{array}$ whereas ${\left(1001\right)}^{1\text{/}3}=10.00332222839093....$ Two terms would suffice for three-digit accuracy.
In the following exercises, use the binomial approximation $\sqrt{1-x}\approx 1-\frac{x}{2}-\frac{{x}^{2}}{8}-\frac{{x}^{3}}{16}-\frac{5{x}^{4}}{128}-\frac{7{x}^{5}}{256}$ for $\left|x\right|<1$ to approximate each number. Compare this value to the value given by a scientific calculator.
[T] $\frac{1}{\sqrt{2}}$ using $x=\frac{1}{2}$ in ${\left(1-x\right)}^{1\text{/}2}$
[T] $\sqrt{5}=5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\frac{1}{\sqrt{5}}$ using $x=\frac{4}{5}$ in ${\left(1-x\right)}^{1\text{/}2}$
The approximation is $2.3152;$ the CAS value is $2.23\text{\u2026}.$
[T] $\sqrt{3}=\frac{3}{\sqrt{3}}$ using $x=\frac{2}{3}$ in ${\left(1-x\right)}^{1\text{/}2}$
[T] $\sqrt{6}$ using $x=\frac{5}{6}$ in ${\left(1-x\right)}^{1\text{/}2}$
The approximation is $2.583\text{\u2026};$ the CAS value is $2.449\text{\u2026}.$
Integrate the binomial approximation of $\sqrt{1-x}$ to find an approximation of ${\int}_{0}^{x}\sqrt{1-t}}dt.$
[T] Recall that the graph of $\sqrt{1-{x}^{2}}$ is an upper semicircle of radius $1.$ Integrate the binomial approximation of $\sqrt{1-{x}^{2}}$ up to order $8$ from $x=\mathrm{-1}$ to $x=1$ to estimate $\frac{\pi}{2}.$
$\sqrt{1-{x}^{2}}=1-\frac{{x}^{2}}{2}-\frac{{x}^{4}}{8}-\frac{{x}^{6}}{16}-\frac{5{x}^{8}}{128}+\text{\cdots}.$ Thus
${\int}_{\mathrm{-1}}^{1}\sqrt{1-{x}^{2}}}dx=x-\frac{{x}^{3}}{6}-\frac{{x}^{5}}{40}-\frac{{x}^{7}}{7\xb716}-\frac{5{x}^{9}}{9\xb7128}+\text{\cdots}{|}_{\mathrm{-1}}^{1}\approx 2-\frac{1}{3}-\frac{1}{20}-\frac{1}{56}-\frac{10}{9\xb7128}+\text{error}=1.590...$ whereas $\frac{\pi}{2}=1.570...$
In the following exercises, use the expansion ${\left(1+x\right)}^{1\text{/}3}=1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\text{\cdots}$ to write the first five terms (not necessarily a quartic polynomial) of each expression.
${\left(1+4x\right)}^{1\text{/}3};a=0$
${\left(1+4x\right)}^{4\text{/}3};a=0$
${\left(1+x\right)}^{4\text{/}3}=\left(1+x\right)\left(1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\text{\cdots}\right)=1+\frac{4x}{3}+\frac{2{x}^{2}}{9}-\frac{4{x}^{3}}{81}+\frac{5{x}^{4}}{243}+\text{\cdots}$
${\left(3+2x\right)}^{1\text{/}3};a=\mathrm{-1}$
${\left({x}^{2}+6x+10\right)}^{1\text{/}3};a=\mathrm{-3}$
${\left(1+{\left(x+3\right)}^{2}\right)}^{1\text{/}3}=1+\frac{1}{3}{\left(x+3\right)}^{2}-\frac{1}{9}{\left(x+3\right)}^{4}+\frac{5}{81}{\left(x+3\right)}^{6}-\frac{10}{243}{\left(x+3\right)}^{8}+\text{\cdots}$
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