# 5.4 Comparison tests  (Page 5/7)

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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 $p=2.$

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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
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