6.2 Properties of power series

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• 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 $f\left(x\right)+g\left(x\right).$

Questions & Answers

why are some countries producing inside the ppf
Claire Reply
prove or disprove that balance of trade of trade deficit is a cause of an abnormal demand curve?
Chioma Reply
what's the fixed cost at output zero
Saidou Reply
fixed cost stay the same regardless of the level of output
Luka
what are the differences between change in demand and change in quantity demand
Sulaiman Reply
what is consumers behaviour
Marfo Reply
importance of income
Emmanuel Reply
Tfor settlement of debt. For purchases. For payment of bills. For daily transactions. For social & recreational enjoyment. For business purposes etc
Oyetunde
thanks
Emmanuel
For investment purposes For security purposes For purpose of forecasting & strategizing.
Oyetunde
what is the real definition of economics
jegede Reply
Economics is the study of the use and allocation of (scarce) resources
demsurf
Jegede, what is the "non" real definition of economics then?
Ernest
Economics is a study of how human use limited resources to fulfil their unlimited want
Musa
the study of how a society use scarce factors of production efficiently so as meet aggregate social demand
Marc
what is oligopoly?
Sailo
Oligopoly can be defines as a market where by there is only tmo or more sellers of a commodity
Paamat
Sory not tmo but two
Paamat
incidence of production there is a choice do you agree? justify
Oduro Reply
What is incidence of production? do u mean incidence of tax?
Aryeetey
I want to know about Richard lipsey and robin as the economist and their definition proposed by them
Musa Reply
what are the causes of scarcity And what are the goal scarcity
Musa
scarcity only exist because human wants are unlimited...if human just know how to be contented then scarcity will not exist
Ylaine
what is ment by possibility curve
Ruzaiq
define accounting?teatly
Ahmed Reply
Is the recording, classifying, interpreting record of all transaction
Yuusuf
is still the act of measuring, interpreting and communicating of financial issues
Yuusuf
measuring business or individual finance
Zeyi
Accounting is the process of collecting,recording,classifying,summarizing and interpreting/presenting financial data to the stakeholders for their economic decision making
asri
hi
Otilina
hi
AVIAH
wat is PPC
ALLAJI
what are the different between need and wants
Musa Reply
the major difference is necessity
Yuusuf
explain any four tool of monetary policy to solve the problem of inflation.
Alicesha Reply
bank rate,open market operation,legal reserve requirement
Johnson
what's marginal utility?
Abena Reply
the additional utility you get if you can consume one more unit of the good x
Luka
Thanks... then what's the law of diminishing marginal utility ?
Abena
The utility decreases with every unit you consume (most of the time). The first unit of consumption will therefore give you the highest utility. Sorry about my english
Luka
Okay... I understand now
Abena
Great!
Luka
hello room
Lawal
one of the leading industrial nations of the world ranking second in manufacturing output after the USA is a. Russia b. Germany c. Britain d. Japan
Lawal
china
Siddharth
japan
Siddharth
good morning
Lamin
hi
Rafiu
hi
nivedha
japan
Ylaine
morning
Adegboye
no other questions?
Ylaine
hii
Dipun
I am from India
Dipun
same question are not mentioned
Dipun
first you give my answer
Dipun
hi
adelakun
welcome
Ahmed
dipun naik
Ahmed
whats your question
adelakun
whats your question
adelakun
I am from India
Dipun
retype the questions
adelakun
marginal untility is the last point desire of a consumer that gets benefit from related good/ service.
Saboor
Why are some countries rich and why are some countries poor? . is poorness a human cause?
Yacquub
well several factors are included...it's not just because of human..
Ylaine
what is a correct reason
Vijay
Japan
Lawal
countries which are rich they are developed countries they have good resources minerals technology power knowledge to use the resources poor countries are under developing countries they have lack of resources, knowledge and if they have these so they dont know the use of these resources.
Siddharth
so these knowledgeable people move /migrate to the other rich/developed countries
Siddharth
Poverty of a country is also related to cultural, economical, and military domination. Usually, the dominant country imposes all of these powers when diplomatically needed or sometimes by force.
Ernest
You can also have considerable poverty in a rich country when such poverty is measured within sectors of its population. In other words, economic indicators can sometime mask such poverty.
Ernest
For example, the U.S.A. has a very high measure of GDP per capital, but millions of Americans ( a considerable amount are children) live in poverty.
Ernest
So poverty is not an easy social phenomenon to pin down neatly into one social realm or another.
Ernest
pls what is price ceiling
jasmine
its the max price a seller can charge for a product, mostly imposed by the government to protect the consumer
Luka
its the max price a seller can charge for a product, mostly imposed by the government to protect the consumer plus it must be imposed below the equilibrium price in order to be effective. A shortage will also be created after its imposition.
Zafar
can happiness be measured?
Ylaine
Happiness is too subjective to be measured as an economic phenomenon or reality. I think that happiness happens at several levels of the human condition: biological, psychological, intellectual and at the level of the soul. How can economic theory be scientific about it?
Ernest
about I have read of something called gross happiness index.
Ylaine
Germany
Arthur
what's Neo classical definition of economic
Mohammed
hi
ALLAJI
economic is a social science studied as a relationship between end and needs scarce which have alternative uses
ALLAJI
what's equilibrium
Daniel
What is economies of scale
Jeremiah Reply
In microeconomics, economies of scale are the sum of total costs saved or that a firm has advantage over its competitors due to its scale of operations. More specifically, it is the firm's cost savings per unit of output that it gains as its production increases in scale.
Ernest
one of the leading industrial nations of the world ranking second in manufacturing output after the USA is ......... a. Russia b. Germany c. Britain d. Japan
Lawal
what is supply of demand?
Joseph Reply
supply of demand?
Yuusuf
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?
Abdul Reply
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

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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