# 5.4 Comparison tests  (Page 4/7)

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

$\sum _{n=1}^{\infty }\frac{1}{2n-1}$

Diverges by comparison with harmonic series, since $2n-1\ge n.$

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

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

${a}_{n}=1\text{/}\left(n+1\right)\left(n+2\right)<1\text{/}{n}^{2}.$ Converges by comparison with p -series, $p=2.$

$\sum _{n=1}^{\infty }\frac{1}{n\text{!}}$

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

$\text{sin}\left(1\text{/}n\right)\le 1\text{/}n,$ so converges by comparison with p -series, $p=2.$

$\sum _{n=1}^{\infty }\frac{{\text{sin}}^{2}n}{{n}^{2}}$

$\sum _{n=1}^{\infty }\frac{\text{sin}\left(1\text{/}n\right)}{\sqrt{n}}$

$\text{sin}\left(1\text{/}n\right)\le 1,$ so converges by comparison with p -series, $p=3\text{/}2.$

$\sum _{n=1}^{\infty }\frac{{n}^{1.2}-1}{{n}^{2.3}+1}$

$\sum _{n=1}^{\infty }\frac{\sqrt{n+1}-\sqrt{n}}{n}$

Since $\sqrt{n+1}-\sqrt{n}=1\text{/}\left(\sqrt{n+1}+\sqrt{n}\right)\le 2\text{/}\sqrt{n},$ series converges by comparison with p -series for $p=1.5.$

$\sum _{n=1}^{\infty }\frac{\sqrt{n}}{\sqrt{{n}^{4}+{n}^{2}}}$

Use the limit comparison test to determine whether each of the following series converges or diverges.

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

Converges by limit comparison with p -series for $p>1.$

$\sum _{n=1}^{\infty }{\left(\frac{\text{ln}\phantom{\rule{0.1em}{0ex}}n}{{n}^{0.6}}\right)}^{2}$

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

Converges by limit comparison with p -series, $p=2.$

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

$\sum _{n=1}^{\infty }\frac{1}{{4}^{n}-{3}^{n}}$

Converges by limit comparison with ${4}^{\text{−}n}.$

$\sum _{n=1}^{\infty }\frac{1}{{n}^{2}-n\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}n}$

$\sum _{n=1}^{\infty }\frac{1}{{e}^{\left(1.1\right)n}-{3}^{n}}$

Converges by limit comparison with $1\text{/}{e}^{1.1n}.$

$\sum _{n=1}^{\infty }\frac{1}{{e}^{\left(1.01\right)n}-{3}^{n}}$

$\sum _{n=1}^{\infty }\frac{1}{{n}^{1+1\text{/}n}}$

Diverges by limit comparison with harmonic series.

$\sum _{n=1}^{\infty }\frac{1}{{2}^{1+1\text{/}n}{n}^{1+1\text{/}n}}$

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

Converges by limit comparison with p -series, $p=3.$

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

$\sum _{n=1}^{\infty }\frac{1}{n}\left({\text{tan}}^{-1}n-\frac{\pi }{2}\right)$

Converges by limit comparison with p -series, $p=3.$

$\sum _{n=1}^{\infty }{\left(1-\frac{1}{n}\right)}^{n.n}$ ( Hint: ${\left(1-\frac{1}{n}\right)}^{n}\to 1\text{/}e.\right)$

$\sum _{n=1}^{\infty }\left(1-{e}^{-1\text{/}n}\right)$ ( Hint: $1\text{/}e\approx {\left(1-1\text{/}n\right)}^{n},$ so $1-{e}^{-1\text{/}n}\approx 1\text{/}n.\right)$

Diverges by limit comparison with $1\text{/}n.$

Does $\sum _{n=2}^{\infty }\frac{1}{{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}^{p}}$ converge if $p$ is large enough? If so, for which $p\text{?}$

Does $\sum _{n=1}^{\infty }{\left(\frac{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}{n}\right)}^{p}$ converge if $p$ is large enough? If so, for which $p\text{?}$

Converges for $p>1$ by comparison with a $p$ series for slightly smaller $p.$

For which $p$ does the series $\sum _{n=1}^{\infty }{2}^{pn}\text{/}{3}^{n}$ converge?

For which $p>0$ does the series $\sum _{n=1}^{\infty }\frac{{n}^{p}}{{2}^{n}}$ converge?

Converges for all $p>0.$

For which $r>0$ does the series $\sum _{n=1}^{\infty }\frac{{r}^{{n}^{2}}}{{2}^{n}}$ converge?

For which $r>0$ does the series $\sum _{n=1}^{\infty }\frac{{2}^{n}}{{r}^{{n}^{2}}}$ converge?

Converges for all $r>1.$ If $r>1$ then ${r}^{n}>4,$ say, once $n>\text{ln}\left(2\right)\text{/}\text{ln}\left(r\right)$ and then the series converges by limit comparison with a geometric series with ratio $1\text{/}2.$

Find all values of $p$ and $q$ such that $\sum _{n=1}^{\infty }\frac{{n}^{p}}{{\left(n\text{!}\right)}^{q}}$ converges.

Does $\sum _{n=1}^{\infty }\frac{{\text{sin}}^{2}\left(nr\text{/}2\right)}{n}$ converge or diverge? Explain.

The numerator is equal to $1$ when $n$ is odd and $0$ when $n$ is even, so the series can be rewritten $\sum _{n=1}^{\infty }\frac{1}{2n+1},$ which diverges by limit comparison with the harmonic series.

Explain why, for each $n,$ at least one of $\left\{|\text{sin}\phantom{\rule{0.1em}{0ex}}n|,|\text{sin}\left(n+1\right)|\text{,...},|\text{sin}\phantom{\rule{0.1em}{0ex}}n+6|\right\}$ is larger than $1\text{/}2.$ Use this relation to test convergence of $\sum _{n=1}^{\infty }\frac{|\text{sin}\phantom{\rule{0.1em}{0ex}}n|}{\sqrt{n}}.$

Suppose that ${a}_{n}\ge 0$ and ${b}_{n}\ge 0$ and that $\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ and $\sum _{n=1}^{\infty }{b}^{2}{}_{n}$ converge. Prove that $\sum _{n=1}^{\infty }{a}_{n}{b}_{n}$ converges and $\sum _{n=1}^{\infty }{a}_{n}\phantom{\rule{0.2em}{0ex}}{b}_{n}\le \frac{1}{2}\left(\sum _{n=1}^{\infty }{a}_{n}^{2}+\sum _{n=1}^{\infty }{b}_{n}^{2}\right).$

${\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}$ or ${a}^{2}+{b}^{2}\ge 2ab,$ so convergence follows from comparison of $2{a}_{n}{b}_{n}$ with ${a}^{2}{}_{n}+{b}^{2}{}_{n}.$ Since the partial sums on the left are bounded by those on the right, the inequality holds for the infinite series.

Does $\sum _{n=1}^{\infty }{2}^{\text{−}\text{ln}\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}n}$ converge? ( Hint: Write ${2}^{\text{ln}\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}n}$ as a power of $\text{ln}\phantom{\rule{0.1em}{0ex}}n.\right)$

Does $\sum _{n=1}^{\infty }{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}^{\text{−}\text{ln}\phantom{\rule{0.1em}{0ex}}n}$ converge? ( Hint: Use $t={e}^{\text{ln}\left(t\right)}$ to compare to a $p-\text{series}\text{.}\right)$

${\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}^{\text{−}\text{ln}\phantom{\rule{0.1em}{0ex}}n}={e}^{\text{−}\text{ln}\left(n\right)\text{ln}\phantom{\rule{0.1em}{0ex}}\text{ln}\left(n\right)}.$ If $n$ is sufficiently large, then $\text{ln}\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}n>2,$ so ${\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}^{\text{−}\text{ln}\phantom{\rule{0.1em}{0ex}}n}<1\text{/}{n}^{2},$ and the series converges by comparison to a $p-\text{series}\text{.}$

Does $\sum _{n=2}^{\infty }{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}n\right)}^{\text{−}\text{ln}\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}n}$ converge? ( Hint: Compare ${a}_{n}$ to $1\text{/}n.\right)$

Show that if ${a}_{n}\ge 0$ and $\sum _{n=1}^{\infty }{a}_{n}$ converges, then $\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges. If $\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges, does $\sum _{n=1}^{\infty }{a}_{n}$ necessarily converge?

${a}_{n}\to 0,$ so ${a}^{2}{}_{n}\le |{a}_{n}|$ for large $n.$ Convergence follows from limit comparison. $\sum 1\text{/}{n}^{2}$ converges, but $\sum 1\text{/}n$ does not, so the fact that $\sum _{n=1}^{\infty }{a}^{2}{}_{n}$ converges does not imply that $\sum _{n=1}^{\infty }{a}_{n}$ converges.

#### Questions & Answers

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
characteristics of micro business
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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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?
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