# 6.4 Working with taylor series  (Page 7/11)

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

Use the approximation ${\left(1-x\right)}^{2\text{/}3}=1-\frac{2x}{3}-\frac{{x}^{2}}{9}-\frac{4{x}^{3}}{81}-\frac{7{x}^{4}}{243}-\frac{14{x}^{5}}{729}+\text{⋯}$ for $|x|<1$ to approximate ${2}^{1\text{/}3}={2.2}^{-2\text{/}3}.$

Twice the approximation is $1.260\text{…}$ whereas ${2}^{1\text{/}3}=1.2599....$

Find the $25\text{th}$ derivative of $f\left(x\right)={\left(1+{x}^{2}\right)}^{13}$ at $x=0.$

Find the $99$ th derivative of $f\left(x\right)={\left(1+{x}^{4}\right)}^{25}.$

${f}^{\left(99\right)}\left(0\right)=0$

In the following exercises, find the Maclaurin series of each function.

$f\left(x\right)=x{e}^{2x}$

$f\left(x\right)={2}^{x}$

$\sum _{n=0}^{\infty }\frac{{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}\left(2\right)x\right)}^{n}}{n\text{!}}$

$f\left(x\right)=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}x}{x}$

$f\left(x\right)=\frac{\text{sin}\left(\sqrt{x}\right)}{\sqrt{x}},\phantom{\rule{0.5em}{0ex}}\left(x>0\right),$

For $x>0,\text{sin}\left(\sqrt{x}\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{\left(2n+1\right)\text{/}2}}{\sqrt{x}\left(2n+1\right)\text{!}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{\left(2n+1\right)\text{!}}.$

$f\left(x\right)=\text{sin}\left({x}^{2}\right)$

$f\left(x\right)={e}^{{x}^{3}}$

${e}^{{x}^{3}}=\sum _{n=0}^{\infty }\frac{{x}^{3n}}{n\text{!}}$

$f\left(x\right)={\text{cos}}^{2}x$ using the identity ${\text{cos}}^{2}x=\frac{1}{2}+\frac{1}{2}\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(2x\right)$

$f\left(x\right)={\text{sin}}^{2}x$ using the identity ${\text{sin}}^{2}x=\frac{1}{2}-\frac{1}{2}\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(2x\right)$

${\text{sin}}^{2}x=\text{−}\sum _{k=1}^{\infty }\frac{{\left(-1\right)}^{k}{2}^{2k-1}{x}^{2k}}{\left(2k\right)\text{!}}$

In the following exercises, find the Maclaurin series of $F\left(x\right)={\int }_{0}^{x}f\left(t\right)\phantom{\rule{0.1em}{0ex}}dt$ by integrating the Maclaurin series of $f$ term by term. If $f$ is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.

$F\left(x\right)={\int }_{0}^{x}{e}^{\text{−}{t}^{2}}dt;f\left(t\right)={e}^{\text{−}{t}^{2}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{n\text{!}}$

$F\left(x\right)={\text{tan}}^{-1}x;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\frac{1}{1+{t}^{2}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{t}^{2n}$

${\text{tan}}^{-1}x=\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^{k}{x}^{2k+1}}{2k+1}$

$F\left(x\right)={\text{tanh}}^{-1}x;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\frac{1}{1-{t}^{2}}=\sum _{n=0}^{\infty }{t}^{2n}$

$F\left(x\right)={\text{sin}}^{-1}x;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\frac{1}{\sqrt{1-{t}^{2}}}=\sum _{k=0}^{\infty }\left(\begin{array}{c}\frac{1}{2}\hfill \\ k\hfill \end{array}\right)\frac{{t}^{2k}}{k\text{!}}$

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

$F\left(x\right)={\int }_{0}^{x}\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}dt;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{t}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{\left(2n+1\right)\text{!}}$

$F\left(x\right)={\int }_{0}^{x}\text{cos}\left(\sqrt{t}\right)\phantom{\rule{0.1em}{0ex}}dt;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n}}{\left(2n\right)\text{!}}$

$F\left(x\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{x}^{n+1}}{\left(n+1\right)\left(2n\right)\text{!}}$

$F\left(x\right)={\int }_{0}^{x}\frac{1-\text{cos}\phantom{\rule{0.1em}{0ex}}t}{{t}^{2}}dt;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\frac{1-\text{cos}\phantom{\rule{0.1em}{0ex}}t}{{t}^{2}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{2n}}{\left(2n+2\right)\text{!}}$

$F\left(x\right)={\int }_{0}^{x}\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+t\right)}{t}dt;\phantom{\rule{0.5em}{0ex}}f\left(t\right)=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}\frac{{t}^{n}}{n+1}$

$F\left(x\right)=\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{{x}^{n}}{{n}^{2}}$

In the following exercises, compute at least the first three nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f.$

$f\left(x\right)=\text{sin}\left(x+\frac{\pi }{4}\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cos}\left(\frac{\pi }{4}\right)+\text{cos}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{sin}\left(\frac{\pi }{4}\right)$

$f\left(x\right)=\text{tan}\phantom{\rule{0.1em}{0ex}}x$

$x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\text{⋯}$

$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(\text{cos}\phantom{\rule{0.1em}{0ex}}x\right)$

$f\left(x\right)={e}^{x}\text{cos}\phantom{\rule{0.1em}{0ex}}x$

$1+x-\frac{{x}^{3}}{3}-\frac{{x}^{4}}{6}+\text{⋯}$

$f\left(x\right)={e}^{\text{sin}\phantom{\rule{0.1em}{0ex}}x}$

$f\left(x\right)={\text{sec}}^{2}x$

$1+{x}^{2}+\frac{2{x}^{4}}{3}+\frac{17{x}^{6}}{45}+\text{⋯}$

$f\left(x\right)=\text{tanh}\phantom{\rule{0.1em}{0ex}}x$

$f\left(x\right)=\frac{\text{tan}\sqrt{x}}{\sqrt{x}}$ (see expansion for $\text{tan}\phantom{\rule{0.1em}{0ex}}x\right)$

Using the expansion for $\text{tan}\phantom{\rule{0.1em}{0ex}}x$ gives $1+\frac{x}{3}+\frac{2{x}^{2}}{15}.$

In the following exercises, find the radius of convergence of the Maclaurin series of each function.

$\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+x\right)$

$\frac{1}{1+{x}^{2}}$

$\frac{1}{1+{x}^{2}}=\sum _{n=0}^{\infty }{\left(-1\right)}^{n}{x}^{2n}$ so $R=1$ by the ratio test.

${\text{tan}}^{-1}x$

$\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+{x}^{2}\right)$

$\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+{x}^{2}\right)=\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n-1}}{n}{x}^{2n}$ so $R=1$ by the ratio test.

Find the Maclaurin series of $\text{sinh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}-{e}^{\text{−}x}}{2}.$

Find the Maclaurin series of $\text{cosh}\phantom{\rule{0.1em}{0ex}}x=\frac{{e}^{x}+{e}^{\text{−}x}}{2}.$

Add series of ${e}^{x}$ and ${e}^{\text{−}x}$ term by term. Odd terms cancel and $\text{cosh}\phantom{\rule{0.1em}{0ex}}x=\sum _{n=0}^{\infty }\frac{{x}^{2n}}{\left(2n\right)\text{!}}.$

Differentiate term by term the Maclaurin series of $\text{sinh}\phantom{\rule{0.1em}{0ex}}x$ and compare the result with the Maclaurin series of $\text{cosh}\phantom{\rule{0.1em}{0ex}}x.$

[T] Let ${S}_{n}\left(x\right)=\sum _{k=0}^{n}{\left(-1\right)}^{k}\frac{{x}^{2k+1}}{\left(2k+1\right)\text{!}}$ and ${C}_{n}\left(x\right)=\sum _{n=0}^{n}{\left(-1\right)}^{k}\frac{{x}^{2k}}{\left(2k\right)\text{!}}$ denote the respective Maclaurin polynomials of degree $2n+1$ of $\text{sin}\phantom{\rule{0.1em}{0ex}}x$ and degree $2n$ of $\text{cos}\phantom{\rule{0.1em}{0ex}}x.$ Plot the errors $\frac{{S}_{n}\left(x\right)}{{C}_{n}\left(x\right)}-\text{tan}\phantom{\rule{0.1em}{0ex}}x$ for $n=1,..,5$ and compare them to $x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\frac{17{x}^{7}}{315}-\text{tan}\phantom{\rule{0.1em}{0ex}}x$ on $\left(-\frac{\pi }{4},\frac{\pi }{4}\right).$

The ratio $\frac{{S}_{n}\left(x\right)}{{C}_{n}\left(x\right)}$ approximates $\text{tan}\phantom{\rule{0.1em}{0ex}}x$ better than does ${p}_{7}\left(x\right)=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\frac{17{x}^{7}}{315}$ for $N\ge 3.$ The dashed curves are $\frac{{S}_{n}}{{C}_{n}}-\text{tan}$ for $n=1,2.$ The dotted curve corresponds to $n=3,$ and the dash-dotted curve corresponds to $n=4.$ The solid curve is ${p}_{7}-\text{tan}\phantom{\rule{0.1em}{0ex}}x.$

Use the identity $2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x=\text{sin}\left(2x\right)$ to find the power series expansion of ${\text{sin}}^{2}x$ at $x=0.$ ( Hint: Integrate the Maclaurin series of $\text{sin}\left(2x\right)$ term by term.)

If $y=\sum _{n=0}^{\infty }{a}_{n}{x}^{n},$ find the power series expansions of $x{y}^{\prime }$ and ${x}^{2}y\text{″}.$

By the term-by-term differentiation theorem, ${y}^{\prime }=\sum _{n=1}^{\infty }n{a}_{n}{x}^{n-1}$ so ${y}^{\prime }=\sum _{n=1}^{\infty }n{a}_{n}{x}^{n-1}\phantom{\rule{0.2em}{0ex}}x{y}^{\prime }=\sum _{n=1}^{\infty }n{a}_{n}{x}^{n},$ whereas ${y}^{\prime }=\sum _{n=2}^{\infty }n\left(n-1\right){a}_{n}{x}^{n-2}$ so $xy\text{″}=\sum _{n=2}^{\infty }n\left(n-1\right){a}_{n}{x}^{n}.$

[T] Suppose that $y=\sum _{k=0}^{\infty }{a}_{k}{x}^{k}$ satisfies ${y}^{\prime }=-2xy$ and $y\left(0\right)=0.$ Show that ${a}_{2k+1}=0$ for all $k$ and that ${a}_{2k+2}=\frac{\text{−}{a}_{2k}}{k+1}.$ Plot the partial sum ${S}_{20}$ of $y$ on the interval $\left[-4,4\right].$

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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 .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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it is a goid question and i want to know the answer as well
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what is fullerene does it is used to make bukky balls
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s.
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Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
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how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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