<< Chapter < Page Chapter >> Page >

Rearranging series

Use the fact that

1 1 2 + 1 3 1 4 + 1 5 = ln 2

to rearrange the terms in the alternating harmonic series so the sum of the rearranged series is 3 ln ( 2 ) / 2 .

Let

n = 1 a n = 1 1 2 + 1 3 1 4 + 1 5 1 6 + 1 7 1 8 + .

Since n = 1 a n = ln ( 2 ) , by the algebraic properties of convergent series,

n = 1 1 2 a n = 1 2 1 4 + 1 6 1 8 + = 1 2 n = 1 a n = ln 2 2 .

Now introduce the series n = 1 b n such that for all n 1 , b 2 n 1 = 0 and b 2 n = a n / 2 . Then

n = 1 b n = 0 + 1 2 + 0 1 4 + 0 + 1 6 + 0 1 8 + = ln 2 2 .

Then using the algebraic limit properties of convergent series, since n = 1 a n and n = 1 b n converge, the series n = 1 ( a n + b n ) converges and

n = 1 ( a n + b n ) = n = 1 a n + n = 1 b n = ln 2 + ln 2 2 = 3 ln 2 2 .

Now adding the corresponding terms, a n and b n , we see that

n = 1 ( a n + b n ) = ( 1 + 0 ) + ( 1 2 + 1 2 ) + ( 1 3 + 0 ) + ( 1 4 1 4 ) + ( 1 5 + 0 ) + ( 1 6 + 1 6 ) + ( 1 7 + 0 ) + ( 1 8 1 8 ) + = 1 + 1 3 1 2 + 1 5 + 1 7 1 4 + .

We notice that the series on the right side of the equal sign is a rearrangement of the alternating harmonic series. Since n = 1 ( a n + b n ) = 3 ln ( 2 ) / 2 , we conclude that

1 + 1 3 1 2 + 1 5 + 1 7 1 4 + = 3 ln ( 2 ) 2 .

Therefore, we have found a rearrangement of the alternating harmonic series having the desired property.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Key concepts

  • For an alternating series n = 1 ( −1 ) n + 1 b n , if b k + 1 b k for all k and b k 0 as k , the alternating series converges.
  • If n = 1 | a n | converges, then n = 1 a n converges.

Key equations

  • Alternating series
    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

State whether each of the following series converges absolutely, conditionally, or not at all.

n = 1 ( −1 ) n + 1 n n + 3

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n + 1 n + 3

Does not converge by divergence test. Terms do not tend to zero.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 1 n + 3

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n + 3 n

Converges conditionally by alternating series test, since n + 3 / n is decreasing. Does not converge absolutely by comparison with p -series, p = 1 / 2 .

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 1 n !

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 3 n n !

Converges absolutely by limit comparison to 3 n / 4 n , for example.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( n 1 n ) n

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( n + 1 n ) n

Diverges by divergence test since lim n | a n | = e .

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 sin 2 n

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 cos 2 n

Does not converge. Terms do not tend to zero.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 sin 2 ( 1 / n )

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 cos 2 ( 1 / n )

lim n cos 2 ( 1 / n ) = 1 . Diverges by divergence test.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ln ( 1 / n )

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ln ( 1 + 1 n )

Converges by alternating series test.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n 2 1 + n 4

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n e 1 + n π

Converges conditionally by alternating series test. Does not converge absolutely by limit comparison with p -series, p = π e

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 2 1 / n

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n 1 / n

Diverges; terms do not tend to zero.

Got questions? Get instant answers now!

n = 1 ( −1 ) n ( 1 n 1 / n ) ( Hint: n 1 / n 1 + ln ( n ) / n for large n . )

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n ( 1 cos ( 1 n ) ) ( Hint: cos ( 1 / n ) 1 1 / n 2 for large n . )

Converges by alternating series test. Does not converge absolutely by limit comparison with harmonic series.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( n + 1 n ) ( Hint: Rationalize the numerator.)

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( 1 n 1 n + 1 ) ( Hint: Cross-multiply then rationalize numerator.)

Converges absolutely by limit comparison with p -series, p = 3 / 2 , after applying the hint.

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( ln ( n + 1 ) ln n )

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 n ( tan −1 ( n + 1 ) tan −1 n ) ( Hint: Use Mean Value Theorem.)

Converges by alternating series test since n ( tan −1 ( n + 1 ) tan −1 n ) is decreasing to zero for large n . Does not converge absolutely by limit comparison with harmonic series after applying hint.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

n = 1 ( −1 ) n + 1 ( 1 n 1 n + 1 )

Converges absolutely, since a n = 1 n 1 n + 1 are terms of a telescoping series.

Got questions? Get instant answers now!

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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
Berger describes sociologists as concerned with
Mueller Reply
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 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