<< Chapter < Page Chapter >> Page >

Integrate the power series ln ( 1 + x ) = n = 1 ( −1 ) n + 1 x n n term-by-term to evaluate ln ( 1 + x ) d x .

n = 2 ( −1 ) n x n n ( n 1 )

Got questions? Get instant answers now!

Up to this point, we have shown several techniques for finding power series representations for functions. However, how do we know that these power series are unique? That is, given a function f and a power series for f at a , is it possible that there is a different power series for f at a that we could have found if we had used a different technique? The answer to this question is no. This fact should not seem surprising if we think of power series as polynomials with an infinite number of terms. Intuitively, if

c 0 + c 1 x + c 2 x 2 + = d 0 + d 1 x + d 2 x 2 +

for all values x in some open interval I about zero, then the coefficients c n should equal d n for n 0 . We now state this result formally in [link] .

Uniqueness of power series

Let n = 0 c n ( x a ) n and n = 0 d n ( x a ) n be two convergent power series such that

n = 0 c n ( x a ) n = n = 0 d n ( x a ) n

for all x in an open interval containing a . Then c n = d n for all n 0 .

Proof

Let

f ( x ) = c 0 + c 1 ( x a ) + c 2 ( x a ) 2 + c 3 ( x a ) 3 + = d 0 + d 1 ( x a ) + d 2 ( x a ) 2 + d 3 ( x a ) 3 + .

Then f ( a ) = c 0 = d 0 . By [link] , we can differentiate both series term-by-term. Therefore,

f ( x ) = c 1 + 2 c 2 ( x a ) + 3 c 3 ( x a ) 2 + = d 1 + 2 d 2 ( x a ) + 3 d 3 ( x a ) 2 + ,

and thus, f ( a ) = c 1 = d 1 . Similarly,

f ( x ) = 2 c 2 + 3 · 2 c 3 ( x a ) + = 2 d 2 + 3 · 2 d 3 ( x a ) +

implies that f ( a ) = 2 c 2 = 2 d 2 , and therefore, c 2 = d 2 . More generally, for any integer n 0 , f ( n ) ( a ) = n ! c n = n ! d n , and consequently, c n = d n for all n 0 .

In this section we have shown how to find power series representations for certain functions using various algebraic operations, differentiation, or integration. At this point, however, we are still limited as to the functions for which we can find power series representations. Next, we show how to find power series representations for many more functions by introducing Taylor series.

Key concepts

  • Given two power series n = 0 c n x n and n = 0 d n x n that converge to functions f and g on a common interval I , the sum and difference of the two series converge to f ± g , respectively, on I . In addition, for any real number b and integer m 0 , the series n = 0 b x m c n x n converges to b x m f ( x ) and the series n = 0 c n ( b x m ) n converges to f ( b x m ) whenever bx m is in the interval I .
  • Given two power series that converge on an interval ( R , R ) , the Cauchy product of the two power series converges on the interval ( R , R ) .
  • Given a power series that converges to a function f on an interval ( R , R ) , the series can be differentiated term-by-term and the resulting series converges to f on ( R , R ) . The series can also be integrated term-by-term and the resulting series converges to f ( x ) d x on ( R , R ) .

If f ( x ) = n = 0 x n n ! and g ( x ) = n = 0 ( −1 ) n x n n ! , find the power series of 1 2 ( f ( x ) + g ( x ) ) and of 1 2 ( f ( x ) g ( x ) ) .

1 2 ( f ( x ) + g ( x ) ) = n = 0 x 2 n ( 2 n ) ! and 1 2 ( f ( x ) g ( x ) ) = n = 0 x 2 n + 1 ( 2 n + 1 ) ! .

Got questions? Get instant answers now!

If C ( x ) = n = 0 x 2 n ( 2 n ) ! and S ( x ) = n = 0 x 2 n + 1 ( 2 n + 1 ) ! , find the power series of C ( x ) + S ( x ) and of C ( x ) S ( x ) .

Got questions? Get instant answers now!

In the following exercises, use partial fractions to find the power series of each function.

4 ( x 3 ) ( x + 1 )

4 ( x 3 ) ( x + 1 ) = 1 x 3 1 x + 1 = 1 3 ( 1 x 3 ) 1 1 ( x ) = 1 3 n = 0 ( x 3 ) n n = 0 ( −1 ) n x n = n = 0 ( ( −1 ) n + 1 1 3 n + 1 ) x n

Got questions? Get instant answers now!

5 ( x 2 + 4 ) ( x 2 1 )

5 ( x 2 + 4 ) ( x 2 1 ) = 1 x 2 1 1 4 1 1 + ( x 2 ) 2 = n = 0 x 2 n 1 4 n = 0 ( −1 ) n ( x 2 ) n = n = 0 ( ( −1 ) + ( −1 ) n + 1 1 2 n + 2 ) x 2 n

Got questions? Get instant answers now!

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 2' conversation and receive update notifications?

Ask