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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where is the latest information on a no technology how can I find it
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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?
Practice Key Terms 4

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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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