if and only if
$\left|x\right|<1,$ we conclude that the interval of convergence for the binomial series is
$\left(\mathrm{-1},1\right).$ The behavior at the endpoints depends on
$r.$ It can be shown that for
$r\ge 0$ the series converges at both endpoints; for
$\mathrm{-1}<r<0,$ the series converges at
$x=1$ and diverges at
$x=\mathrm{-1};$ and for
$r<\mathrm{-1},$ the series diverges at both endpoints. The binomial series does converge to
${\left(1+x\right)}^{r}$ in
$\left(\mathrm{-1},1\right)$ for all real numbers
$r,$ but proving this fact by showing that the remainder
${R}_{n}\left(x\right)\to 0$ is difficult.
Definition
For any real number
$r,$ the Maclaurin series for
$f\left(x\right)={\left(1+x\right)}^{r}$ is the binomial series. It converges to
$f$ for
$\left|x\right|<1,$ and we write
We can use this definition to find the binomial series for
$f\left(x\right)=\sqrt{1+x}$ and use the series to approximate
$\sqrt{1.5}.$
Finding binomial series
Find the binomial series for
$f\left(x\right)=\sqrt{1+x}.$
Use the third-order Maclaurin polynomial
${p}_{3}\left(x\right)$ to estimate
$\sqrt{1.5}.$ Use Taylor’s theorem to bound the error. Use a graphing utility to compare the graphs of
$f$ and
${p}_{3}.$
Here
$r=\frac{1}{2}.$ Using the definition for the binomial series, we obtain
for some
$c$ between
$0$ and
$0.5.$ Since
${f}^{\left(4\right)}\left(x\right)=-\frac{15}{{2}^{4}{\left(1+x\right)}^{7\text{/}2}},$ and the maximum value of
$\left|{f}^{\left(4\right)}\left(x\right)\right|$ on the interval
$\left(0,0.5\right)$ occurs at
$x=0,$ we have
At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form
$f\left(x\right)={\left(1+x\right)}^{r}.$ In
[link] , we summarize the results of these series. We remark that the convergence of the Maclaurin series for
$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(1+x\right)$ at the endpoint
$x=1$ and the Maclaurin series for
$f\left(x\right)={\text{tan}}^{\mathrm{-1}}x$ at the endpoints
$x=1$ and
$x=\mathrm{-1}$ relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.)
Earlier in the chapter, we showed how you could combine power series to create new power series. Here we use these properties, combined with the Maclaurin series in
[link] , to create Maclaurin series for other functions.
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?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
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
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?