<< Chapter < Page Chapter >> Page >

Limit comparison test

Let a n , b n 0 for all n 1 .

  1. If lim n a n / b n = L 0 , then n = 1 a n and n = 1 b n both converge or both diverge.
  2. If lim n a n / b n = 0 and n = 1 b n converges, then n = 1 a n converges.
  3. If lim n a n / b n = and n = 1 b n diverges, then n = 1 a n diverges.

Note that if a n / b n 0 and n = 1 b n diverges, the limit comparison test gives no information. Similarly, if a n / b n and n = 1 b n converges, the test also provides no information. For example, consider the two series n = 1 1 / n and n = 1 1 / n 2 . These series are both p -series with p = 1 / 2 and p = 2 , respectively. Since p = 1 / 2 > 1 , the series n = 1 1 / n diverges. On the other hand, since p = 2 < 1 , the series n = 1 1 / n 2 converges. However, suppose we attempted to apply the limit comparison test, using the convergent p series n = 1 1 / n 3 as our comparison series. First, we see that

1 / n 1 / n 3 = n 3 n = n 5 / 2 as n .

Similarly, we see that

1 / n 2 1 / n 3 = n as n .

Therefore, if a n / b n when n = 1 b n converges, we do not gain any information on the convergence or divergence of n = 1 a n .

Using the limit comparison test

For each of the following series, use the limit comparison test to determine whether the series converges or diverges. If the test does not apply, say so.

  1. n = 1 1 n + 1
  2. n = 1 2 n + 1 3 n
  3. n = 1 ln ( n ) n 2
  1. Compare this series to n = 1 1 n . Calculate
    lim n 1 / ( n + 1 ) 1 / n = lim n n n + 1 = lim n 1 / n 1 + 1 / n = 1 .
    By the limit comparison test, since n = 1 1 n diverges, then n = 1 1 n + 1 diverges.
  2. Compare this series to n = 1 ( 2 3 ) n . We see that
    lim n ( 2 n + 1 ) / 3 n 2 n / 3 n = lim n 2 n + 1 3 n · 3 n 2 n = lim n 2 n + 1 2 n = lim n [ 1 + ( 1 2 ) n ] = 1 .

    Therefore,
    lim n ( 2 n + 1 ) / 3 n 2 n / 3 n = 1 .

    Since n = 1 ( 2 3 ) n converges, we conclude that n = 1 2 n + 1 3 n converges.
  3. Since ln n < n , compare with n = 1 1 n . We see that
    lim n ln n / n 2 1 / n = lim n ln n n 2 · n 1 = lim n ln n n .

    In order to evaluate lim n ln n / n , evaluate the limit as x of the real-valued function ln ( x ) / x . These two limits are equal, and making this change allows us to use L’Hôpital’s rule. We obtain
    lim x ln x x = lim x 1 x = 0 .

    Therefore, lim n ln n / n = 0 , and, consequently,
    lim n ln n / n 2 1 / n = 0 .

    Since the limit is 0 but n = 1 1 n diverges, the limit comparison test does not provide any information.
    Compare with n = 1 1 n 2 instead. In this case,
    lim n ln n / n 2 1 / n 2 = lim n ln n n 2 · n 2 1 = lim n ln n = .

    Since the limit is but n = 1 1 n 2 converges, the test still does not provide any information.
    So now we try a series between the two we already tried. Choosing the series n = 1 1 n 3 / 2 , we see that
    lim n ln n / n 2 1 / n 3 / 2 = lim n ln n n 2 · n 3 / 2 1 = lim n ln n n .

    As above, in order to evaluate lim n ln n / n , evaluate the limit as x of the real-valued function ln x / x . Using L’Hôpital’s rule,
    lim x ln x x = lim x 2 x x = lim x 2 x = 0 .

    Since the limit is 0 and n = 1 1 n 3 / 2 converges, we can conclude that n = 1 ln n n 2 converges.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use the limit comparison test to determine whether the series n = 1 5 n 3 n + 2 converges or diverges.

The series diverges.

Got questions? Get instant answers now!

Key concepts

  • The comparison tests are used to determine convergence or divergence of series with positive terms.
  • When using the comparison tests, a series n = 1 a n is often compared to a geometric or p -series.

Use the comparison test to determine whether the following series converge.

n = 1 a n where a n = 2 n ( n + 1 )

Got questions? Get instant answers now!

n = 1 a n where a n = 1 n ( n + 1 / 2 )

Converges by comparison with 1 / n 2 .

Got questions? Get instant answers now!

Questions & Answers

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
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
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
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how to synthesize TiO2 nanoparticles by chemical methods
Zubear
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 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