# 5.4 Comparison tests  (Page 5/7)

 Page 5 / 7

Suppose that ${a}_{n}>0$ for all $n$ and that $\sum _{n=1}^{\infty }{a}_{n}$ converges. Suppose that ${b}_{n}$ is an arbitrary sequence of zeros and ones. Does $\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ necessarily converge?

Suppose that ${a}_{n}>0$ for all $n$ and that $\sum _{n=1}^{\infty }{a}_{n}$ diverges. Suppose that ${b}_{n}$ is an arbitrary sequence of zeros and ones with infinitely many terms equal to one. Does $\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ necessarily diverge?

No. $\sum _{n=1}^{\infty }1\text{/}n$ diverges. Let ${b}_{k}=0$ unless $k={n}^{2}$ for some $n.$ Then $\sum _{k}{b}_{k}\text{/}k=\sum 1\text{/}{k}^{2}$ converges.

Complete the details of the following argument: If $\sum _{n=1}^{\infty }\frac{1}{n}$ converges to a finite sum $s,$ then $\frac{1}{2}s=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\text{⋯}$ and $s-\frac{1}{2}s=1+\frac{1}{3}+\frac{1}{5}+\text{⋯}.$ Why does this lead to a contradiction?

Show that if ${a}_{n}\ge 0$ and $\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges, then $\sum _{n=1}^{\infty }{\text{sin}}^{2}\left({a}_{n}\right)$ converges.

$|\text{sin}\phantom{\rule{0.1em}{0ex}}t|\le |t|,$ so the result follows from the comparison test.

Suppose that ${a}_{n}\text{/}{b}_{n}\to 0$ in the comparison test, where ${a}_{n}\ge 0$ and ${b}_{n}\ge 0.$ Prove that if $\sum {b}_{n}$ converges, then $\sum {a}_{n}$ converges.

Let ${b}_{n}$ be an infinite sequence of zeros and ones. What is the largest possible value of $x=\sum _{n=1}^{\infty }{b}_{n}\text{/}{2}^{n}\text{?}$

By the comparison test, $x=\sum _{n=1}^{\infty }{b}_{n}\text{/}{2}^{n}\le \sum _{n=1}^{\infty }1\text{/}{2}^{n}=1.$

Let ${d}_{n}$ be an infinite sequence of digits, meaning ${d}_{n}$ takes values in $\left\{0,\phantom{\rule{0.2em}{0ex}}1\text{,…},9\right\}.$ What is the largest possible value of $x=\sum _{n=1}^{\infty }{d}_{n}\text{/}{10}^{n}$ that converges?

Explain why, if $x>1\text{/}2,$ then $x$ cannot be written $x=\sum _{n=2}^{\infty }\frac{{b}_{n}}{{2}^{n}}\left({b}_{n}=0\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}1,\phantom{\rule{0.2em}{0ex}}{b}_{1}=0\right).$

If ${b}_{1}=0,$ then, by comparison, $x\le \sum _{n=2}^{\infty }1\text{/}{2}^{n}=1\text{/}2.$

[T] Evelyn has a perfect balancing scale, an unlimited number of $1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights, and one each of $1\text{/}2\phantom{\rule{0.2em}{0ex}}\text{-kg},\phantom{\rule{0.2em}{0ex}}1\text{/}4\phantom{\rule{0.2em}{0ex}}\text{-kg},\phantom{\rule{0.2em}{0ex}}1\text{/}8\phantom{\rule{0.2em}{0ex}}\text{-kg},$ and so on weights. She wishes to weigh a meteorite of unspecified origin to arbitrary precision. Assuming the scale is big enough, can she do it? What does this have to do with infinite series?

[T] Robert wants to know his body mass to arbitrary precision. He has a big balancing scale that works perfectly, an unlimited collection of $1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights, and nine each of $0.1\phantom{\rule{0.2em}{0ex}}\text{-kg,}$ $0.01\phantom{\rule{0.2em}{0ex}}\text{-kg},\phantom{\rule{0.2em}{0ex}}0.001\phantom{\rule{0.2em}{0ex}}\text{-kg,}$ and so on weights. Assuming the scale is big enough, can he do this? What does this have to do with infinite series?

Yes. Keep adding $1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights until the balance tips to the side with the weights. If it balances perfectly, with Robert standing on the other side, stop. Otherwise, remove one of the $1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights, and add $0.1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights one at a time. If it balances after adding some of these, stop. Otherwise if it tips to the weights, remove the last $0.1\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weight. Start adding $0.01\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights. If it balances, stop. If it tips to the side with the weights, remove the last $0.01\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weight that was added. Continue in this way for the $0.001\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights, and so on. After a finite number of steps, one has a finite series of the form $A+\sum _{n=1}^{N}{s}_{n}\text{/}{10}^{n}$ where $A$ is the number of full kg weights and ${d}_{n}$ is the number of $1\text{/}{10}^{n}\phantom{\rule{0.2em}{0ex}}\text{-kg}$ weights that were added. If at some state this series is Robert’s exact weight, the process will stop. Otherwise it represents the $N\text{th}$ partial sum of an infinite series that gives Robert’s exact weight, and the error of this sum is at most $1\text{/}{10}^{N}.$

The series $\sum _{n=1}^{\infty }\frac{1}{2n}$ is half the harmonic series and hence diverges. It is obtained from the harmonic series by deleting all terms in which $n$ is odd. Let $m>1$ be fixed. Show, more generally, that deleting all terms $1\text{/}n$ where $n=mk$ for some integer $k$ also results in a divergent series.

In view of the previous exercise, it may be surprising that a subseries of the harmonic series in which about one in every five terms is deleted might converge. A depleted harmonic series is a series obtained from $\sum _{n=1}^{\infty }\frac{1}{n}$ by removing any term $1\text{/}n$ if a given digit, say $9,$ appears in the decimal expansion of $n.$ Argue that this depleted harmonic series converges by answering the following questions.

1. How many whole numbers $n$ have $d$ digits?
2. How many $d\text{-digit}$ whole numbers $h\left(d\right).$ do not contain $9$ as one or more of their digits?
3. What is the smallest $d\text{-digit}$ number $m\left(d\right)\text{?}$
4. Explain why the deleted harmonic series is bounded by $\sum _{d=1}^{\infty }\frac{h\left(d\right)}{m\left(d\right)}.$
5. Show that $\sum _{d=1}^{\infty }\frac{h\left(d\right)}{m\left(d\right)}$ converges.

a. ${10}^{d}-{10}^{d-1}<{10}^{d}$ b. $h\left(d\right)<{9}^{d}$ c. $m\left(d\right)={10}^{d-1}+1$ d. Group the terms in the deleted harmonic series together by number of digits. $h\left(d\right)$ bounds the number of terms, and each term is at most $1\text{/}m\left(d\right).$ $\sum _{d=1}^{\infty }h\left(d\right)\text{/}m\left(d\right)\le \sum _{d=1}^{\infty }{9}^{d}\text{/}{\left(10\right)}^{d-1}\le 90.$ One can actually use comparison to estimate the value to smaller than $80.$ The actual value is smaller than $23.$

Suppose that a sequence of numbers ${a}_{n}>0$ has the property that ${a}_{1}=1$ and ${a}_{n+1}=\frac{1}{n+1}{S}_{n},$ where ${S}_{n}={a}_{1}+\text{⋯}+{a}_{n}.$ Can you determine whether $\sum _{n=1}^{\infty }{a}_{n}$ converges? ( Hint: ${S}_{n}$ is monotone.)

Suppose that a sequence of numbers ${a}_{n}>0$ has the property that ${a}_{1}=1$ and ${a}_{n+1}=\frac{1}{{\left(n+1\right)}^{2}}{S}_{n},$ where ${S}_{n}={a}_{1}+\text{⋯}+{a}_{n}.$ Can you determine whether $\sum _{n=1}^{\infty }{a}_{n}$ converges? ( Hint: ${S}_{2}={a}_{2}+{a}_{1}={a}_{2}+{S}_{1}={a}_{2}+1=1+1\text{/}4=\left(1+1\text{/}4\right){S}_{1},$ ${S}_{3}=\frac{1}{{3}^{2}}{S}_{2}+{S}_{2}=\left(1+1\text{/}9\right){S}_{2}=\left(1+1\text{/}9\right)\left(1+1\text{/}4\right){S}_{1},$ etc. Look at $\text{ln}\left({S}_{n}\right),$ and use $\text{ln}\left(1+t\right)\le t,$ $t>0.\right)$

Continuing the hint gives ${S}_{N}=\left(1+1\text{/}{N}^{2}\right)\left(1+1\text{/}{\left(N-1\right)}^{2}\text{…}\left(1+1\text{/}4\right)\right).$ Then $\text{ln}\left({S}_{N}\right)=\text{ln}\left(1+1\text{/}{N}^{2}\right)+\text{ln}\left(1+1\text{/}{\left(N-1\right)}^{2}\right)+\text{⋯}+\text{ln}\left(1+1\text{/}4\right).$ Since $\text{ln}\left(1+t\right)$ is bounded by a constant times $t,$ when $0 one has $\text{ln}\left({S}_{N}\right)\le C\sum _{n=1}^{N}\frac{1}{{n}^{2}},$ which converges by comparison to the p -series for $p=2.$

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Difference between extinct and extici spicies
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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 By Rhodes By Tess Armstrong By OpenStax By OpenStax By Sandhills MLT By Janet Forrester By Hoy Wen By OpenStax By Katie Montrose By Rohini Ajay