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We remark that this theorem is true more generally as long as there exists some integer N such that 0 b n + 1 b n for all n N .

Convergence of alternating series

For each of the following alternating series, determine whether the series converges or diverges.

  1. n = 1 ( −1 ) n + 1 / n 2
  2. n = 1 ( −1 ) n + 1 n / ( n + 1 )
  1. Since
    1 ( n + 1 ) 2 < 1 n 2 and 1 n 2 0 ,
    the series converges.
  2. Since n / ( n + 1 ) 0 as n , we cannot apply the alternating series test. Instead, we use the n th term test for divergence. Since
    lim n ( −1 ) n + 1 n n + 1 0 ,
    the series diverges.
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Determine whether the series n = 1 ( −1 ) n + 1 n / 2 n converges or diverges.

The series converges.

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Remainder of an alternating series

It is difficult to explicitly calculate the sum of most alternating series, so typically the sum is approximated by using a partial sum. When doing so, we are interested in the amount of error in our approximation. Consider an alternating series

n = 1 ( −1 ) n + 1 b n

satisfying the hypotheses of the alternating series test. Let S denote the sum of this series and { S k } be the corresponding sequence of partial sums. From [link] , we see that for any integer N 1 , the remainder R N satisfies

| R N | = | S S N | | S N + 1 S N | = b n + 1 .

Remainders in alternating series

Consider an alternating series of the form

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

that satisfies the hypotheses of the alternating series test. Let S denote the sum of the series and S N denote the N th partial sum. For any integer N 1 , the remainder R N = S S N satisfies

| R N | b N + 1 .

In other words, if the conditions of the alternating series test apply, then the error in approximating the infinite series by the N th partial sum S N is in magnitude at most the size of the next term b N + 1 .

Estimating the remainder of an alternating series

Consider the alternating series

n = 1 ( −1 ) n + 1 n 2 .

Use the remainder estimate to determine a bound on the error R 10 if we approximate the sum of the series by the partial sum S 10 .

From the theorem stated above,

| R 10 | b 11 = 1 11 2 0.008265 .

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Find a bound for R 20 when approximating n = 1 ( −1 ) n + 1 / n by S 20 .


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Absolute and conditional convergence

Consider a series n = 1 a n and the related series n = 1 | a n | . Here we discuss possibilities for the relationship between the convergence of these two series. For example, consider the alternating harmonic series n = 1 ( −1 ) n + 1 / n . The series whose terms are the absolute value of these terms is the harmonic series, since n = 1 | ( −1 ) n + 1 / n | = n = 1 1 / n . Since the alternating harmonic series converges, but the harmonic series diverges, we say the alternating harmonic series exhibits conditional convergence.

By comparison, consider the series n = 1 ( −1 ) n + 1 / n 2 . The series whose terms are the absolute values of the terms of this series is the series n = 1 1 / n 2 . Since both of these series converge, we say the series n = 1 ( −1 ) n + 1 / n 2 exhibits absolute convergence.


A series n = 1 a n exhibits conditional convergence    if n = 1 | a n | converges. A series n = 1 a n exhibits absolute convergence    if n = 1 a n converges but n = 1 | a n | diverges.

As shown by the alternating harmonic series, a series n = 1 a n may converge, but n = 1 | a n | may diverge. In the following theorem, however, we show that if n = 1 | a n | converges, then n = 1 a n converges.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket 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?
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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