# 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 $p=1.5.$

$\sum _{n=1}^{\infty }\frac{\sqrt[4]{n}}{\sqrt[3]{{n}^{4}+{n}^{2}}}$

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.

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
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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?
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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
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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?
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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.
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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