<< Chapter < Page Chapter >> Page >
  • Use the alternating series test to test an alternating series for convergence.
  • Estimate the sum of an alternating series.
  • Explain the meaning of absolute convergence and conditional convergence.

So far in this chapter, we have primarily discussed series with positive terms. In this section we introduce alternating series—those series whose terms alternate in sign. We will show in a later chapter that these series often arise when studying power series. After defining alternating series, we introduce the alternating series test to determine whether such a series converges.

The alternating series test

A series whose terms alternate between positive and negative values is an alternating series    . For example, the series

n = 1 ( 1 2 ) n = 1 2 + 1 4 1 8 + 1 16

and

n = 1 ( −1 ) n + 1 n = 1 1 2 + 1 3 1 4 +

are both alternating series.

Definition

Any series whose terms alternate between positive and negative values is called an alternating series. An alternating series can be written in the form

n = 1 ( −1 ) n + 1 b n = b 1 b 2 + b 3 b 4 +

or

n 1 ( −1 ) n b n = b 1 + b 2 b 3 + b 4

Where b n 0 for all positive integers n .

Series (1), shown in [link] , is a geometric series. Since | r | = | 1 / 2 | < 1 , the series converges. Series (2), shown in [link] , is called the alternating harmonic series. We will show that whereas the harmonic series diverges, the alternating harmonic series converges.

To prove this, we look at the sequence of partial sums { S k } ( [link] ).

Proof

Consider the odd terms S 2 k + 1 for k 0 . Since 1 / ( 2 k + 1 ) < 1 / 2 k ,

S 2 k + 1 = S 2 k 1 1 2 k + 1 2 k + 1 < S 2 k 1 .

Therefore, { S 2 k + 1 } is a decreasing sequence. Also,

S 2 k + 1 = ( 1 1 2 ) + ( 1 3 1 4 ) + + ( 1 2 k 1 1 2 k ) + 1 2 k + 1 > 0 .

Therefore, { S 2 k + 1 } is bounded below. Since { S 2 k + 1 } is a decreasing sequence that is bounded below, by the Monotone Convergence Theorem, { S 2 k + 1 } converges. Similarly, the even terms { S 2 k } form an increasing sequence that is bounded above because

S 2 k = S 2 k 2 + 1 2 k 1 1 2 k > S 2 k 2

and

S 2 k = 1 + ( 1 2 + 1 3 ) + + ( 1 2 k 2 + 1 2 k 1 ) 1 2 k < 1 .

Therefore, by the Monotone Convergence Theorem, the sequence { S 2 k } also converges. Since

S 2 k + 1 = S 2 k + 1 2 k + 1 ,

we know that

lim k S 2 k + 1 = lim k S 2 k + lim k 1 2 k + 1 .

Letting S = lim k S 2 k + 1 and using the fact that 1 / ( 2 k + 1 ) 0 , we conclude that lim k S 2 k = S . Since the odd terms and the even terms in the sequence of partial sums converge to the same limit S , it can be shown that the sequence of partial sums converges to S , and therefore the alternating harmonic series converges to S .

It can also be shown that S = ln 2 , and we can write

n = 1 ( −1 ) n + 1 n = 1 1 2 + 1 3 1 4 + = ln ( 2 ) .
This graph demonstrates the alternating hamanic series in the first quadrant. The highest line 1 is drawn to S1, the next line -1/2 is drawn to S2, the next line +1/3 is drawn to S3, the line -1/4 is drawn to S4, and the last line +1/5 is drawn to S5. The odd terms are decreasing and bounded below, and the even terms are increasing and bounded above. It seems to be converging to S, which is in the middle of S2, S4 and S5, S3, S1.
For the alternating harmonic series, the odd terms S 2 k + 1 in the sequence of partial sums are decreasing and bounded below. The even terms S 2 k are increasing and bounded above.

More generally, any alternating series of form (3) ( [link] ) or (4) ( [link] ) converges as long as b 1 b 2 b 3 and b n 0 ( [link] ). The proof is similar to the proof for the alternating harmonic series.

This diagram illustrates an alternating series in quadrant 1. The highest line b1 is drawn out to S1, the next line –b2 is drawn back to S2, the next line b3 is drawn out to S3, the next line –b4 is drawn back to S4, and the last line is drawn out to S5. It seems to be converging to S, which is in between S2, S4 and S5, S3, and S1. The odd terms are decreasing and bounded below. The even terms are increasing and bounded above.
For an alternating series b 1 b 2 + b 3 in which b 1 > b 2 > b 3 > , the odd terms S 2 k + 1 in the sequence of partial sums are decreasing and bounded below. The even terms S 2 k are increasing and bounded above.

Alternating series test

An alternating series of the form

n = 1 ( −1 ) n + 1 b n or n = 1 ( −1 ) n b n

converges if

  1. 0 b n + 1 b n for all n 1 and
  2. lim n b n = 0 .

This is known as the alternating series test    .

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
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 4

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