# 6.2 Properties of power series

 Page 1 / 10
• Combine power series by addition or subtraction.
• Create a new power series by multiplication by a power of the variable or a constant, or by substitution.
• Multiply two power series together.
• Differentiate and integrate power series term-by-term.

In the preceding section on power series and functions we showed how to represent certain functions using power series. In this section we discuss how power series can be combined, differentiated, or integrated to create new power series. This capability is particularly useful for a couple of reasons. First, it allows us to find power series representations for certain elementary functions, by writing those functions in terms of functions with known power series. For example, given the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find a power series representation for ${f}^{\prime }\left(x\right)=\frac{1}{{\left(1-x\right)}^{2}}.$ Second, being able to create power series allows us to define new functions that cannot be written in terms of elementary functions. This capability is particularly useful for solving differential equations for which there is no solution in terms of elementary functions.

## Combining power series

If we have two power series with the same interval of convergence, we can add or subtract the two series to create a new power series, also with the same interval of convergence. Similarly, we can multiply a power series by a power of x or evaluate a power series at ${x}^{m}$ for a positive integer m to create a new power series. Being able to do this allows us to find power series representations for certain functions by using power series representations of other functions. For example, since we know the power series representation for $f\left(x\right)=\frac{1}{1-x},$ we can find power series representations for related functions, such as

$y=\frac{3x}{1-{x}^{2}}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}y=\frac{1}{\left(x-1\right)\left(x-3\right)}.$

In [link] we state results regarding addition or subtraction of power series, composition of a power series, and multiplication of a power series by a power of the variable. For simplicity, we state the theorem for power series centered at $x=0.$ Similar results hold for power series centered at $x=a.$

## Combining power series

Suppose that the two power series $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n}$ converge to the functions f and g , respectively, on a common interval I .

1. The power series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}±{d}_{n}{x}^{n}\right)$ converges to $f±g$ on I .
2. For any integer $m\ge 0$ and any real number b , the power series $\sum _{n=0}^{\infty }b{x}^{m}{c}_{n}{x}^{n}$ converges to $b{x}^{m}f\left(x\right)$ on I .
3. For any integer $m\ge 0$ and any real number b , the series $\sum _{n=0}^{\infty }{c}_{n}{\left(b{x}^{m}\right)}^{n}$ converges to $f\left(b{x}^{m}\right)$ for all x such that $b{x}^{m}$ is in I .

## Proof

We prove i. in the case of the series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right).$ Suppose that $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n}$ converge to the functions f and g , respectively, on the interval I . Let x be a point in I and let ${S}_{N}\left(x\right)$ and ${T}_{N}\left(x\right)$ denote the N th partial sums of the series $\sum _{n=0}^{\infty }{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty }{d}_{n}{x}^{n},$ respectively. Then the sequence $\left\{{S}_{N}\left(x\right)\right\}$ converges to $f\left(x\right)$ and the sequence $\left\{{T}_{N}\left(x\right)\right\}$ converges to $g\left(x\right).$ Furthermore, the N th partial sum of $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)$ is

$\begin{array}{cc}\hfill \sum _{n=0}^{N}\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)& =\sum _{n=0}^{N}{c}_{n}{x}^{n}+\sum _{n=0}^{N}{d}_{n}{x}^{n}\hfill \\ & ={S}_{N}\left(x\right)+{T}_{N}\left(x\right).\hfill \end{array}$

Because

$\begin{array}{cc}\hfill \underset{N\to \infty }{\text{lim}}\left({S}_{N}\left(x\right)+{T}_{N}\left(x\right)\right)& =\underset{N\to \infty }{\text{lim}}{S}_{N}\left(x\right)+\underset{N\to \infty }{\text{lim}}{T}_{N}\left(x\right)\hfill \\ & =f\left(x\right)+g\left(x\right),\hfill \end{array}$

we conclude that the series $\sum _{n=0}^{\infty }\left({c}_{n}{x}^{n}+{d}_{n}{x}^{n}\right)$ converges to $f\left(x\right)+g\left(x\right).$

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
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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
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 By By   By  By   By Lakeima Roberts