6.4 Working with taylor series  (Page 6/11)

 Page 6 / 11

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.

Key concepts

• The binomial series is the Maclaurin series for $f\left(x\right)={\left(1+x\right)}^{r}.$ It converges for $|x|<1.$
• Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
• Power series can be used to solve differential equations.
• Taylor series can be used to help approximate integrals that cannot be evaluated by other means.

In the following exercises, use appropriate substitutions to write down the Maclaurin series for the given binomial.

${\left(1-x\right)}^{1\text{/}3}$

${\left(1+{x}^{2}\right)}^{-1\text{/}3}$

${\left(1+{x}^{2}\right)}^{-1\text{/}3}=\sum _{n=0}^{\infty }\left(\begin{array}{c}-\frac{1}{3}\hfill \\ \hfill n\hfill \end{array}\right){x}^{2n}$

${\left(1-x\right)}^{1.01}$

${\left(1-2x\right)}^{2\text{/}3}$

${\left(1-2x\right)}^{2\text{/}3}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{2}^{n}\left(\begin{array}{c}\frac{2}{3}\hfill \\ n\hfill \end{array}\right){x}^{n}$

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 $a=0$

$\sqrt{2+{x}^{2}}=\sum _{n=0}^{\infty }{2}^{\left(1\text{/}2\right)-n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){x}^{2n};\left(|{x}^{2}|<2\right)$

$\sqrt{x+2}$ at $a=1$

$\sqrt{2x-{x}^{2}}$ at $a=1$ ( Hint: $2x-{x}^{2}=1-{\left(x-1\right)}^{2}\right)$

$\sqrt{2x-{x}^{2}}=\sqrt{1-{\left(x-1\right)}^{2}}$ so $\sqrt{2x-{x}^{2}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-1\right)}^{2n}$

${\left(x-8\right)}^{1\text{/}3}$ at $a=9$

$\sqrt{x}$ at $a=4$

$\sqrt{x}=2\sqrt{1+\frac{x-4}{4}}$ so $\sqrt{x}=\sum _{n=0}^{\infty }{2}^{1-2n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-4\right)}^{n}$

${x}^{1\text{/}3}$ at $a=27$

$\sqrt{x}$ at $x=9$

$\sqrt{x}=\sum _{n=0}^{\infty }{3}^{1-3n}\left(\begin{array}{c}\frac{1}{2}\hfill \\ n\hfill \end{array}\right){\left(x-9\right)}^{n}$

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 ${\left(1000+x\right)}^{1\text{/}3}$

$10{\left(1+\frac{x}{1000}\right)}^{1\text{/}3}=\sum _{n=0}^{\infty }{10}^{1-3n}\left(\begin{array}{c}\frac{1}{3}\hfill \\ n\hfill \end{array}\right){x}^{n}.$ Using, for example, a fourth-degree estimate at $x=1$ gives $\begin{array}{cc}\hfill {\left(1001\right)}^{1\text{/}3}& \approx 10\left(1+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 1\hfill \end{array}\right){10}^{-3}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 2\hfill \end{array}\right){10}^{-6}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 3\hfill \end{array}\right){10}^{-9}+\left(\begin{array}{c}\frac{1}{3}\hfill \\ 4\hfill \end{array}\right){10}^{-12}\right)\hfill \\ & =10\left(1+\frac{1}{{3.10}^{3}}-\frac{1}{{9.10}^{6}}+\frac{5}{{81.10}^{9}}-\frac{10}{{243.10}^{12}}\right)=10.00333222...\hfill \end{array}$ 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 $|x|<1$ 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] $\sqrt{5}=5\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\frac{1}{\sqrt{5}}$ using $x=\frac{4}{5}$ in ${\left(1-x\right)}^{1\text{/}2}$

The approximation is $2.3152;$ the CAS value is $2.23\text{…}.$

[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 $2.583\text{…};$ the CAS value is $2.449\text{…}.$

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 $x=-1$ to $x=1$ to estimate $\frac{\pi }{2}.$

$\sqrt{1-{x}^{2}}=1-\frac{{x}^{2}}{2}-\frac{{x}^{4}}{8}-\frac{{x}^{6}}{16}-\frac{5{x}^{8}}{128}+\text{⋯}.$ Thus

${\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx=x-\frac{{x}^{3}}{6}-\frac{{x}^{5}}{40}-\frac{{x}^{7}}{7·16}-\frac{5{x}^{9}}{9·128}+\text{⋯}{|}_{-1}^{1}\approx 2-\frac{1}{3}-\frac{1}{20}-\frac{1}{56}-\frac{10}{9·128}+\text{error}=1.590...$ whereas $\frac{\pi }{2}=1.570...$

In the following exercises, use the expansion ${\left(1+x\right)}^{1\text{/}3}=1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\text{⋯}$ to write the first five terms (not necessarily a quartic polynomial) of each expression.

${\left(1+4x\right)}^{1\text{/}3};a=0$

${\left(1+4x\right)}^{4\text{/}3};a=0$

${\left(1+x\right)}^{4\text{/}3}=\left(1+x\right)\left(1+\frac{1}{3}x-\frac{1}{9}{x}^{2}+\frac{5}{81}{x}^{3}-\frac{10}{243}{x}^{4}+\text{⋯}\right)=1+\frac{4x}{3}+\frac{2{x}^{2}}{9}-\frac{4{x}^{3}}{81}+\frac{5{x}^{4}}{243}+\text{⋯}$

${\left(3+2x\right)}^{1\text{/}3};a=-1$

${\left({x}^{2}+6x+10\right)}^{1\text{/}3};a=-3$

${\left(1+{\left(x+3\right)}^{2}\right)}^{1\text{/}3}=1+\frac{1}{3}{\left(x+3\right)}^{2}-\frac{1}{9}{\left(x+3\right)}^{4}+\frac{5}{81}{\left(x+3\right)}^{6}-\frac{10}{243}{\left(x+3\right)}^{8}+\text{⋯}$

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
Abdul