# 5.3 The divergence and integral tests

 Page 1 / 9
• Use the divergence test to determine whether a series converges or diverges.
• Use the integral test to determine the convergence of a series.
• Estimate the value of a series by finding bounds on its remainder term.

In the previous section, we determined the convergence or divergence of several series by explicitly calculating the limit of the sequence of partial sums $\left\{{S}_{k}\right\}.$ In practice, explicitly calculating this limit can be difficult or impossible. Luckily, several tests exist that allow us to determine convergence or divergence for many types of series. In this section, we discuss two of these tests: the divergence test and the integral test. We will examine several other tests in the rest of this chapter and then summarize how and when to use them.

## Divergence test

For a series $\sum _{n=1}^{\infty }{a}_{n}$ to converge, the $n\text{th}$ term ${a}_{n}$ must satisfy ${a}_{n}\to 0$ as $n\to \infty .$

Therefore, from the algebraic limit properties of sequences,

$\underset{k\to \infty }{\text{lim}}{a}_{k}=\underset{k\to \infty }{\text{lim}}\left({S}_{k}-{S}_{k-1}\right)=\underset{k\to \infty }{\text{lim}}{S}_{k}-\underset{k\to \infty }{\text{lim}}{S}_{k-1}=S-S=0.$

Therefore, if $\sum _{n=1}^{\infty }{a}_{n}$ converges, the $n\text{th}$ term ${a}_{n}\to 0$ as $n\to \infty .$ An important consequence of this fact is the following statement:

$\text{If}\phantom{\rule{0.2em}{0ex}}{a}_{n}↛0\phantom{\rule{0.2em}{0ex}}\text{as}\phantom{\rule{0.2em}{0ex}}n\to \infty ,\sum _{n=1}^{\infty }{a}_{n}\phantom{\rule{0.2em}{0ex}}\text{diverges}.$

This test is known as the divergence test    because it provides a way of proving that a series diverges.

## Divergence test

If $\underset{n\to \infty }{\text{lim}}{a}_{n}=c\ne 0$ or $\underset{n\to \infty }{\text{lim}}{a}_{n}$ does not exist, then the series $\sum _{n=1}^{\infty }{a}_{n}$ diverges.

It is important to note that the converse of this theorem is not true. That is, if $\underset{n\to \infty }{\text{lim}}{a}_{n}=0,$ we cannot make any conclusion about the convergence of $\sum _{n=1}^{\infty }{a}_{n}.$ For example, $\underset{n\to 0}{\text{lim}}\left(1\text{/}n\right)=0,$ but the harmonic series $\sum _{n=1}^{\infty }1\text{/}n$ diverges. In this section and the remaining sections of this chapter, we show many more examples of such series. Consequently, although we can use the divergence test to show that a series diverges, we cannot use it to prove that a series converges. Specifically, if ${a}_{n}\to 0,$ the divergence test is inconclusive.

## Using the divergence test

For each of the following series, apply the divergence test. If the divergence test proves that the series diverges, state so. Otherwise, indicate that the divergence test is inconclusive.

1. $\sum _{n=1}^{\infty }\frac{n}{3n-1}$
2. $\sum _{n=1}^{\infty }\frac{1}{{n}^{3}}$
3. $\sum _{n=1}^{\infty }{e}^{1\text{/}{n}^{2}}$
1. Since $n\text{/}\left(3n-1\right)\to 1\text{/}3\ne 0,$ by the divergence test, we can conclude that
$\sum _{n=1}^{\infty }\frac{n}{3n-1}$

diverges.
2. Since $1\text{/}{n}^{3}\to 0,$ the divergence test is inconclusive.
3. Since ${e}^{1\text{/}{n}^{2}}\to 1\ne 0,$ by the divergence test, the series
$\sum _{n=1}^{\infty }{e}^{1\text{/}{n}^{2}}$

diverges.

What does the divergence test tell us about the series $\sum _{n=1}^{\infty }\text{cos}\left(1\text{/}{n}^{2}\right)\text{?}$

The series diverges.

## Integral test

In the previous section, we proved that the harmonic series diverges by looking at the sequence of partial sums $\left\{{S}_{k}\right\}$ and showing that ${S}_{{2}^{k}}>1+k\text{/}2$ for all positive integers $k.$ In this section we use a different technique to prove the divergence of the harmonic series. This technique is important because it is used to prove the divergence or convergence of many other series. This test, called the integral test    , compares an infinite sum to an improper integral. It is important to note that this test can only be applied when we are considering a series whose terms are all positive.

To illustrate how the integral test works, use the harmonic series as an example. In [link] , we depict the harmonic series by sketching a sequence of rectangles with areas $1,1\text{/}2,1\text{/}3,1\text{/}4\text{,…}$ along with the function $f\left(x\right)=1\text{/}x.$ From the graph, we see that

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
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.
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?
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