<< Chapter < Page Chapter >> Page >
n = 0 ( r n ) x n = 1 + r x + r ( r 1 ) 2 ! x 2 + + r ( r 1 ) ( r n + 1 ) n ! x n + .

We now need to determine the interval of convergence for the binomial series [link] . We apply the ratio test. Consequently, we consider

| a n + 1 | | a n | = | r ( r 1 ) ( r 2 ) ( r n ) | x | | n + 1 ( n + 1 ) ! · n | r ( r 1 ) ( r 2 ) ( r n + 1 ) | | x | n = | r n | | x | | n + 1 | .

Since

lim n | a n + 1 | | a n | = | x | < 1

if and only if | x | < 1 , we conclude that the interval of convergence for the binomial series is ( −1 , 1 ) . The behavior at the endpoints depends on r . It can be shown that for r 0 the series converges at both endpoints; for −1 < r < 0 , the series converges at x = 1 and diverges at x = −1 ; and for r < −1 , the series diverges at both endpoints. The binomial series does converge to ( 1 + x ) r in ( −1 , 1 ) for all real numbers r , but proving this fact by showing that the remainder R n ( x ) 0 is difficult.

Definition

For any real number r , the Maclaurin series for f ( x ) = ( 1 + x ) r is the binomial series. It converges to f for | x | < 1 , and we write

( 1 + x ) r = n = 0 ( r n ) x n = 1 + r x + r ( r 1 ) 2 ! x 2 + + r ( r 1 ) ( r n + 1 ) n ! x n +

for | x | < 1 .

We can use this definition to find the binomial series for f ( x ) = 1 + x and use the series to approximate 1.5 .

Finding binomial series

  1. Find the binomial series for f ( x ) = 1 + x .
  2. Use the third-order Maclaurin polynomial p 3 ( x ) to estimate 1.5 . Use Taylor’s theorem to bound the error. Use a graphing utility to compare the graphs of f and p 3 .
  1. Here r = 1 2 . Using the definition for the binomial series, we obtain
    1 + x = 1 + 1 2 x + ( 1 / 2 ) ( 1 / 2 ) 2 ! x 2 + ( 1 / 2 ) ( 1 / 2 ) ( 3 / 2 ) 3 ! x 3 + = 1 + 1 2 x 1 2 ! 1 2 2 x 2 + 1 3 ! 1 · 3 2 3 x 3 + ( −1 ) n + 1 n ! 1 · 3 · 5 ( 2 n 3 ) 2 n x n + = 1 + n = 1 ( −1 ) n + 1 n ! 1 · 3 · 5 ( 2 n 3 ) 2 n x n .
  2. From the result in part a. the third-order Maclaurin polynomial is
    p 3 ( x ) = 1 + 1 2 x 1 8 x 2 + 1 16 x 3 .

    Therefore,
    1.5 = 1 + 0.5 1 + 1 2 ( 0.5 ) 1 8 ( 0.5 ) 2 + 1 16 ( 0.5 ) 3 1.2266.

    From Taylor’s theorem, the error satisfies
    R 3 ( 0.5 ) = f ( 4 ) ( c ) 4 ! ( 0.5 ) 4

    for some c between 0 and 0.5 . Since f ( 4 ) ( x ) = 15 2 4 ( 1 + x ) 7 / 2 , and the maximum value of | f ( 4 ) ( x ) | on the interval ( 0 , 0.5 ) occurs at x = 0 , we have
    | R 3 ( 0.5 ) | 15 4 ! 2 4 ( 0.5 ) 4 0.00244 .

    The function and the Maclaurin polynomial p 3 are graphed in [link] .
    This graph has two curves. The first one is f(x)= the square root of (1+x) and the second is psub3(x). The curves are very close at y = 1.
    The third-order Maclaurin polynomial p 3 ( x ) provides a good approximation for f ( x ) = 1 + x for x near zero.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the binomial series for f ( x ) = 1 ( 1 + x ) 2 .

n = 0 ( −1 ) n ( n + 1 ) x n

Got questions? Get instant answers now!

Common functions expressed as taylor series

At this point, we have derived Maclaurin series for exponential, trigonometric, and logarithmic functions, as well as functions of the form f ( x ) = ( 1 + x ) r . In [link] , we summarize the results of these series. We remark that the convergence of the Maclaurin series for f ( x ) = ln ( 1 + x ) at the endpoint x = 1 and the Maclaurin series for f ( x ) = tan −1 x at the endpoints x = 1 and x = −1 relies on a more advanced theorem than we present here. (Refer to Abel’s theorem for a discussion of this more technical point.)

Maclaurin series for common functions
Function Maclaurin Series Interval of Convergence
f ( x ) = 1 1 x n = 0 x n −1 < x < 1
f ( x ) = e x n = 0 x n n ! < x <
f ( x ) = sin x n = 0 ( −1 ) n x 2 n + 1 ( 2 n + 1 ) ! < x <
f ( x ) = cos x n = 0 ( −1 ) n x 2 n ( 2 n ) ! < x <
f ( x ) = ln ( 1 + x ) n = 0 ( −1 ) n + 1 x n n −1 < x 1
f ( x ) = tan −1 x n = 0 ( −1 ) n x 2 n + 1 2 n + 1 −1 < x 1
f ( x ) = ( 1 + x ) r n = 0 ( r n ) x n −1 < x < 1

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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





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