# 5.5 Alternating series  (Page 4/10)

 Page 4 / 10

## Rearranging series

Use the fact that

$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\text{⋯}=\text{ln}\phantom{\rule{0.1em}{0ex}}2$

to rearrange the terms in the alternating harmonic series so the sum of the rearranged series is $3\phantom{\rule{0.1em}{0ex}}\text{ln}\left(2\right)\text{/}2.$

Let

$\sum _{n=1}^{\infty }{a}_{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\text{⋯}.$

Since $\sum _{n=1}^{\infty }{a}_{n}=\text{ln}\left(2\right),$ by the algebraic properties of convergent series,

$\sum _{n=1}^{\infty }\frac{1}{2}{a}_{n}=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\text{⋯}=\frac{1}{2}\sum _{n=1}^{\infty }{a}_{n}=\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}2}{2}.$

Now introduce the series $\sum _{n=1}^{\infty }{b}_{n}$ such that for all $n\ge 1,$ ${b}_{2n-1}=0$ and ${b}_{2n}={a}_{n}\text{/}2.$ Then

$\sum _{n=1}^{\infty }{b}_{n}=0+\frac{1}{2}+0-\frac{1}{4}+0+\frac{1}{6}+0-\frac{1}{8}+\text{⋯}=\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}2}{2}.$

Then using the algebraic limit properties of convergent series, since $\sum _{n=1}^{\infty }{a}_{n}$ and $\sum _{n=1}^{\infty }{b}_{n}$ converge, the series $\sum _{n=1}^{\infty }\left({a}_{n}+{b}_{n}\right)$ converges and

$\sum _{n=1}^{\infty }\left({a}_{n}+{b}_{n}\right)=\sum _{n=1}^{\infty }{a}_{n}+\sum _{n=1}^{\infty }{b}_{n}=\text{ln}\phantom{\rule{0.1em}{0ex}}2+\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}2}{2}=\frac{3\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}2}{2}.$

Now adding the corresponding terms, ${a}_{n}$ and ${b}_{n},$ we see that

$\begin{array}{cc}\hfill \sum _{n=1}^{\infty }\left({a}_{n}+{b}_{n}\right)& =\left(1+0\right)+\left(-\frac{1}{2}+\frac{1}{2}\right)+\left(\frac{1}{3}+0\right)+\left(-\frac{1}{4}-\frac{1}{4}\right)+\left(\frac{1}{5}+0\right)+\left(-\frac{1}{6}+\frac{1}{6}\right)\hfill \\ \\ & \phantom{\rule{1.5em}{0ex}}+\left(\frac{1}{7}+0\right)+\left(\frac{1}{8}-\frac{1}{8}\right)+\text{⋯}\hfill \\ & =1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\text{⋯}.\hfill \end{array}$

We notice that the series on the right side of the equal sign is a rearrangement of the alternating harmonic series. Since $\sum _{n=1}^{\infty }\left({a}_{n}+{b}_{n}\right)=3\phantom{\rule{0.1em}{0ex}}\text{ln}\left(2\right)\text{/}2,$ we conclude that

$1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\frac{1}{4}+\text{⋯}=\frac{3\phantom{\rule{0.1em}{0ex}}\text{ln}\left(2\right)}{2}.$

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

## Key concepts

• For an alternating series $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n},$ if ${b}_{k+1}\le {b}_{k}$ for all $k$ and ${b}_{k}\to 0$ as $k\to \infty ,$ the alternating series converges.
• If $\sum _{n=1}^{\infty }|{a}_{n}|$ converges, then $\sum _{n=1}^{\infty }{a}_{n}$ converges.

## Key equations

• Alternating series
$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\text{⋯}\phantom{\rule{0.2em}{0ex}}\text{or}$
$\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}=\text{−}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\text{⋯}$

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

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{n}{n+3}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{\sqrt{n}+1}{\sqrt{n}+3}$

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

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{\sqrt{n+3}}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{\sqrt{n+3}}{n}$

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

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{n\text{!}}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{{3}^{n}}{n\text{!}}$

Converges absolutely by limit comparison to ${3}^{n}\text{/}{4}^{n},$ for example.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\left(\frac{n-1}{n}\right)}^{n}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\left(\frac{n+1}{n}\right)}^{n}$

Diverges by divergence test since $\underset{n\to \infty }{\text{lim}}|{a}_{n}|=e.$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\text{sin}}^{2}n$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\text{cos}}^{2}n$

Does not converge. Terms do not tend to zero.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\text{sin}}^{2}\left(1\text{/}n\right)$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{\text{cos}}^{2}\left(1\text{/}n\right)$

$\underset{n\to \infty }{\text{lim}}{\text{cos}}^{2}\left(1\text{/}n\right)=1.$ Diverges by divergence test.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{ln}\left(1\text{/}n\right)$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\text{ln}\left(1+\frac{1}{n}\right)$

Converges by alternating series test.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{{n}^{2}}{1+{n}^{4}}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{{n}^{e}}{1+{n}^{\pi }}$

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

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{2}^{1\text{/}n}$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{n}^{1\text{/}n}$

Diverges; terms do not tend to zero.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n}\left(1-{n}^{1\text{/}n}\right)$ ( Hint: ${n}^{1\text{/}n}\approx 1+\text{ln}\left(n\right)\text{/}n$ for large $n.\right)$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}n\left(1-\text{cos}\left(\frac{1}{n}\right)\right)$ ( Hint: $\text{cos}\left(1\text{/}n\right)\approx 1-1\text{/}{n}^{2}$ for large $n.\right)$

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

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\left(\sqrt{n+1}-\sqrt{n}\right)$ ( Hint: Rationalize the numerator.)

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)$ ( Hint: Cross-multiply then rationalize numerator.)

Converges absolutely by limit comparison with p -series, $p=3\text{/}2,$ after applying the hint.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\left(\text{ln}\left(n+1\right)-\text{ln}\phantom{\rule{0.2em}{0ex}}n\right)$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}n\left({\text{tan}}^{-1}\left(n+1\right)-{\text{tan}}^{-1}n\right)$ ( Hint: Use Mean Value Theorem.)

Converges by alternating series test since $n\left({\text{tan}}^{-1}\left(n+1\right)\text{−}{\text{tan}}^{-1}n\right)$ is decreasing to zero for large $n.$ Does not converge absolutely by limit comparison with harmonic series after applying hint.

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\left({\left(n+1\right)}^{2}-{n}^{2}\right)$

$\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\left(\frac{1}{n}-\frac{1}{n+1}\right)$

Converges absolutely, since ${a}_{n}=\frac{1}{n}-\frac{1}{n+1}$ are terms of a telescoping series.

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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
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