# 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.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
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

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