Being able to represent a function by an “infinite polynomial” is a powerful tool. Polynomial functions are the easiest functions to analyze, since they only involve the basic arithmetic operations of addition, subtraction, multiplication, and division. If we can represent a complicated function by an infinite polynomial, we can use the polynomial representation to differentiate or integrate it. In addition, we can use a truncated version of the polynomial expression to approximate values of the function. So, the question is, when can we represent a function by a power series?
converges if and only if
$\left|r\right|<1.$ In that case, it converges to
$\frac{a}{1-r}.$ Therefore, if
$\left|x\right|<1,$ the series in
[link] converges to
$\frac{1}{1-x}$ and we write
We now show graphically how this series provides a representation for the function
$f\left(x\right)=\frac{1}{1-x}$ by comparing the graph of
f with the graphs of several of the partial sums of this infinite series.
Graphing a function and partial sums of its power series
Sketch a graph of
$f\left(x\right)=\frac{1}{1-x}$ and the graphs of the corresponding partial sums
${S}_{N}\left(x\right)={\displaystyle \sum _{n=0}^{N}{x}^{n}}$ for
$N=2,4,6$ on the interval
$\left(\mathrm{-1},1\right).$ Comment on the approximation
${S}_{N}$ as
N increases.
From the graph in
[link] you see that as
N increases,
${S}_{N}$ becomes a better approximation for
$f\left(x\right)=\frac{1}{1-x}$ for
x in the interval
$\left(\mathrm{-1},1\right).$
Sketch a graph of
$f\left(x\right)=\frac{1}{1-{x}^{2}}$ and the corresponding partial sums
${S}_{N}\left(x\right)={\displaystyle \sum _{n=0}^{N}{x}^{2n}}$ for
$N=2,4,6$ on the interval
$\left(\mathrm{-1},1\right).$
Next we consider functions involving an expression similar to the sum of a geometric series and show how to represent these functions using power series.
Representing a function with a power series
Use a power series to represent each of the following functions
$f.$ Find the interval of convergence.
$f\left(x\right)=\frac{1}{1+{x}^{3}}$
$f\left(x\right)=\frac{{x}^{2}}{4-{x}^{2}}$
You should recognize this function
f as the sum of a geometric series, because
Since this series converges if and only if
$\left|\text{\u2212}{x}^{3}\right|<1,$ the interval of convergence is
$\left(\mathrm{-1},1\right),$ and we have
This function is not in the exact form of a sum of a geometric series. However, with a little algebraic manipulation, we can relate
f to a geometric series. By factoring 4 out of the two terms in the denominator, we obtain
The series converges as long as
$\left|{\left(\frac{x}{2}\right)}^{2}\right|<1$ (note that when
$\left|{\left(\frac{x}{2}\right)}^{2}\right|=1$ the series does not converge). Solving this inequality, we conclude that the interval of convergence is
$\left(\mathrm{-2},2\right)$ and
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.
In this morden time nanotechnology used in many field .
1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc
2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc
3- Atomobile -MEMS, Coating on car etc.
and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change .
maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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?