# Convergence of a series

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## Homework Statement

Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or not

## The Attempt at a Solution

looking at ## 1/sin (n) ## by comparison,
##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1##
for ##n≥1 ##
implying that ##{sin (n)}≤n ##
##1/2sin (n) ≤1/{n} ## converges, up to this point i have been trying to look at the trig. term in the series....
again by comparison,...

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## Homework Statement

Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or not

## The Attempt at a Solution

looking at ## 1/sin (n) ## by comparison,
##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1##
for ##n≥1 ##
implying that ##{sin (n)}≤n ##
##1/2sin (n) ≤1/{n} ## converges, up to this point i have been trying to look at the trig. term in the series....
again by comparison,...
I think the first thing you need to do is eliminate the trig term by finding a bounding series. What series ##f(n)## can you come up that is just a little bit larger than ##\frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## and has no trig term?

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## Homework Statement

Determine whether the series ## \frac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}## converges or not

## The Attempt at a Solution

looking at ## 1/sin (n) ## by comparison,
##1/n^2=1+1/4+1/9+1/16+...## converges for ##n≥1##
for ##n≥1 ##
implying that ##{sin (n)}≤n ##
##1/2sin (n) ≤1/{n} ## converges, up to this point i have been trying to look at the trig. term in the series....
again by comparison,...
Look at
$$\text{numerator} = (n^3 +3n)^{1/2} = n^{3/2} \left( 1 + \frac 3 n \right)^{1/2}$$
and
$$\text{denominator} = 5n^3 + 3 n^2 + 2 \sin(n) = 5 n^3 \left( 1 + \frac{3}{5n} + \frac{2 \sin(n)}{5n^3} \right).$$
The numerator is ##< 2 \, n^{3/2}##, and for any small ##\epsilon > 0## we can find ##N> 0## so that ##n > N## implies the denominator is ##> 5n^3 (1 - \epsilon).##

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Delta2 and chwala
Gold Member
Look at
$$\text{numerator} = (n^3 +3n)^{1/2} = n^{3/2} \left( 1 + \frac 3 n \right)^{1/2}$$
and
$$\text{denominator} = 5n^3 + 3 n^2 + 2 \sin(n) = 5 n^3 \left( 1 + \frac{3}{5n} + \frac{2 \sin(n)}{5n^3} \right).$$
The numerator is ##< 2 \, n^{3/2}##, and for any small ##\epsilon > 0## we can find ##N> 0## so that ##n > N## implies the denominator is ##> 5n^3 (1 - \epsilon).##
let me post my attempt..

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there look at my working attached........

#### Attachments

• convergenceproblem.docx
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Gold Member
since the series ## 1/n^3## converges for the p-series p greater than 1, it suffices to say that our series converges though it fails for the limit comparison test....my thoughts

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there look at my working attached........

This attachment shows up on my laptop computer, but opens empty and blank on my i-phone. You should avoid using tools that require the reader to use particular media to read your posts. Avoid WORD; use LaTeX instead. It is faster, easier and looks better. Now is the time for you to start learning it; you can install a complete system on your computer for $0 (for typing up your assignments), and it is built-in in this Forum as well. chwala Gold Member This attachment shows up on my laptop computer, but opens empty and blank on my i-phone. You should avoid using tools that require the reader to use particular media to read your posts. Avoid WORD; use LaTeX instead. It is faster, easier and looks better. Now is the time for you to start learning it; you can install a complete system on your computer for$0 (for typing up your assignments), and it is built-in in this Forum as well.
ok sir, allow me to repost in latex....doing so in a few hours let me finish with a class...

Gold Member
our problem ##\sum_{n=1}^\infty## ##\frac {(n^2+3n)^{1/2}} {5n^3+3n^2+ 2 sin n}\ ##
*give me a few more hours finishing with a class*
##sum_{n=1}^\infty## ##{\frac {3}{n^2}+1}##
GIVE ME TIMEAM IN CLASS

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Homework Helper
Gold Member
I also cant read your work in word document (yeap living in 2018 and haven't installed Microsoft Word :D)…

But , following the hints from @Ray Vickson, the final sequence ##b_n## which we ll use for the direct comparison test is ##b_n=\frac{2n^{3/2}}{5\epsilon n^3}=\frac{2}{5\epsilon}\frac{1}{n^{3/2}}## so your final conclusion should be "because the series of ##b_n## converges (p-series with p=3/2>1) and ##|a_n|<|b_n|## for n>N for some positive ##\epsilon## and ##N##, from direct comparison test it follows that the series of ##a_n## converges."

chwala
Gold Member
I also cant read your work in word document (yeap living in 2018 and haven't installed Microsoft Word :D)…

But , following the hints from @Ray Vickson, the final sequence ##b_n## which we ll use for the direct comparison test is ##b_n=\frac{2n^{3/2}}{5\epsilon n^3}=\frac{2}{5\epsilon}\frac{1}{n^{3/2}}## so your final conclusion should be "because the series of ##b_n## converges (p-series with p=3/2>1) and ##|a_n|<|b_n|## for n>N for some positive ##\epsilon## and ##N##, from direct comparison test it follows that the series of ##a_n## converges."
kindly let's be patient as i am reposting the solution using latex, i am learning a few latex terms.....give me time...

Delta2
Mentor
@chwala, the work you showed in the Word document is wrong right from the start.
You have
$$\sum_{n = 1}^\infty \frac{\sqrt{n^2 + 3n}}{5n^3 + 3n^2 + 2\sin n} = \sum_{n = 1}^\infty \frac {\frac{\sqrt{n(n^2 + 3)}}{n^3}}{5 + \frac 3 n + \frac{2\sin n}{n^3}}$$

In post #1 you have ##(n^3 + 3n)^{1/2}## in the numerator, but above you have ##\sqrt{n^2 + 3n}## on the left side. That's a relatively minor typo, but confusing for someone who might not have read things carefully throughout the thread.

BTW, since you're struggling with LaTeX, here's what I typed for the stuff above, in unrendered form:
\sum_{n = 1}^\infty \frac{\sqrt{n^2 + 3n}}{5n^3 + 3n^2 + 2\sin n} = \sum_{n = 1}^\infty \frac {\frac{\sqrt{n(n^2 + 3)}}{n^3}}{5 + \frac 3 n + \frac{2\sin n}{n^3}}

One thing to keep in mind for exponents, subscripts, fraction parts, limits of integration, and possibly a few more things: if the exponent, fraction part, etc. is just a single character, you don't need to include braces around it.
For example, this is fine: x^2
But here you need braces: x^{-2}

And this is fine \frac 1 2
But here you need braces: \frac {x - 2} 3 and \frac {x - 2}{3x}

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Homework Helper
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kindly let's be patient as i am reposting the solution using latex, i am learning a few latex terms.....give me time...

Nobody is rushing you. If you need a few days, take a few days.

chwala
Gold Member
Nobody is rushing you. If you need a few days, take a few days.
thanks i have been busy now i can embark on physicsforums...

Gold Member
@chwala, the work you showed in the Word document is wrong right from the start.
You have
$$\sum_{n = 1}^\infty \frac{\sqrt{n^2 + 3n}}{5n^3 + 3n^2 + 2\sin n} = \sum_{n = 1}^\infty \frac {\frac{\sqrt{n(n^2 + 3)}}{n^3}}{5 + \frac 3 n + \frac{2\sin n}{n^3}}$$

In post #1 you have ##(n^3 + 3n)^{1/2}## in the numerator, but above you have ##\sqrt{n^2 + 3n}## on the left side. That's a relatively minor typo, but confusing for someone who might not have read things carefully throughout the thread.

BTW, since you're struggling with LaTeX, here's what I typed for the stuff above, in unrendered form:
\sum_{n = 1}^\infty \frac{\sqrt{n^2 + 3n}}{5n^3 + 3n^2 + 2\sin n} = \sum_{n = 1}^\infty \frac {\frac{\sqrt{n(n^2 + 3)}}{n^3}}{5 + \frac 3 n + \frac{2\sin n}{n^3}}

One thing to keep in mind for exponents, subscripts, fraction parts, limits of integration, and possibly a few more things: if the exponent, fraction part, etc. is just a single character, you don't need to include braces around it.
For example, this is fine: x^2
But here you need braces: x^{-2}

And this is fine \frac 1 2
But here you need braces: \frac {x - 2} 3 and \frac {x - 2}{3x}
noted, let me retype my whole solution using latex...

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Mentor
While you're figuring out LaTeX, something you should consider is that
$$\sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2} \le \sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2\sin(n)} \le \sum_{n = 1}^\infty\frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 - 2}$$
If you can show that the outer summations are convergent, you'll be able to say something about the series you're working with.

chwala
Gold Member
am back guys...was on vacation, let me embark on this again...

Gold Member
While you're figuring out LaTeX, something you should consider is that
$$\sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2} \le \sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2\sin(n)} \le \sum_{n = 1}^\infty\frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 - 2}$$
If you can show that the outer summations are convergent, you'll be able to say something about the series you're working with.
bet its easy to show by comparison method.

Gold Member
Allow me to look at this post again, i hope its not closed. I have been busy lately...

Staff Emeritus
Homework Helper
Gold Member
Allow me to look at this post again, i hope its not closed. I have been busy lately...
It looks to me like it's open.

member 587159
Mentor
Allow me to look at this post again, i hope its not closed. I have been busy lately...
Yes, it's open, although 5+ months seems a long time to figure out this problem.

Gold Member
Let me look at this again, i need to refresh on convergence, my apologies

Gold Member
Yes, it's open, although 5+ months seems a long time to figure out this problem.
can you give me more insight Mark? i am stuck in this rumble...

Mentor
From post #9, of July 18:
chwala said:
our problem ##\sum_{n=1}^\infty## ##\frac {(n^2+3n)^{1/2}} {5n^3+3n^2+ 2 sin n}\ ##
The dominant term in the numerator is n, and the dominant term in the denominator is ##5n^3##. Do you know the Limit Comparison Test?

valenumr
Gold Member
ok let me look at this using comparison test

Mentor
ok let me look at this using comparison test
The Limit Comparison Test would be the better, and easier choice.

Gold Member
While you're figuring out LaTeX, something you should consider is that
$$\sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2} \le \sum_{n = 1}^\infty \frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 + 2\sin(n)} \le \sum_{n = 1}^\infty\frac{(n^3 + 3n)^{1/2}}{5n^3 + 3n^2 - 2}$$
If you can show that the outer summations are convergent, you'll be able to say something about the series you're working with.
Mark i want to respond to this thread...its bad of me to not have given it priority...i will respond over the weekend...cheers

Gold Member
From Post ##3## and ##10##, we make use of the comparison test by making use of another similar series to determine whether our series is convergent or not.
We have,
## \dfrac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}##=## \dfrac {n^3(1+\frac{3}{n})^{1/2}} {5n^3(1+\frac{3}{5n}+\frac{2 sin (n)}{5n^3})}##≤##\dfrac {2n^{\frac {3}{2}} }{5εn^3}##=##\dfrac {2}{5εn^{\frac {3}{2}}}##
Using ##p## series, the series converges because ##\dfrac{3}{2}##>##1##. This Implies that our series is convergent.

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Mentor
From Post ##3## and ##10##, we make use of the comparison test by making use of another similar series to determine whether our series is convergent or not.
We have,
## \dfrac {(n^3+3n)^{1/2}} {5n^3+3n^2+2 sin (n)}##=## \dfrac {n^3(1+\frac{3}{n})^{1/2}} {5n^3(1+\frac{3}{5n}+\frac{2 sin (n)}{5n^3})}##≤##\dfrac {2n^{\frac {3}{2}} }{5εn^3}##=##\dfrac {2}{5εn^{\frac {3}{2}}}##
Using ##p## series, the series converges because ##\dfrac{3}{2}##>##1##. This Implies that our series is convergent.
1. Is that ##\epsilon## in the 3rd and 4th expressions a typo? I don't see why it should be there.
2. There's a typo in the 2nd expression. The ##n^3## that you brought out should be ##n^{3/2}##.
3. You're on somewhat shaky ground in saying that the 2nd expression is ##\le## the third expression. If the numerator in the 2nd expression had been only ##n^3## with the same denominator it would be clear that 2nd expression ##\le## your third expression. However, when both numerator and denominator are increasing, it's more difficult to make that case. Instead of the comparison test, I would use the limit comparison test.
For a simpler example of what I'm talking about in #3, consider ##\frac{1 + x}{3 + y}## vs. ##\frac 2 3##. Is the first fraction smaller or larger than the second? It's clear that ##\frac 1 {3 + y} \le \frac 1 3## for positive y, but having both numerator and denominator changing makes it more difficult to prove.

chwala
Gold Member
1. Is that ##\epsilon## in the 3rd and 4th expressions a typo? I don't see why it should be there.
2. There's a typo in the 2nd expression. The ##n^3## that you brought out should be ##n^{3/2}##.
3. You're on somewhat shaky ground in saying that the 2nd expression is ##\le## the third expression. If the numerator in the 2nd expression had been only ##n^3## with the same denominator it would be clear that 2nd expression ##\le## your third expression. However, when both numerator and denominator are increasing, it's more difficult to make that case. Instead of the comparison test, I would use the limit comparison test.
For a simpler example of what I'm talking about in #3, consider ##\frac{1 + x}{3 + y}## vs. ##\frac 2 3##. Is the first fraction smaller or larger than the second? It's clear that ##\frac 1 {3 + y} \le \frac 1 3## for positive y, but having both numerator and denominator changing makes it more difficult to prove.
Thanks Mark, I will use limits and further spend the next two days refreshing on convergence/divergence of sequences and series...cheers...It was epsilon...I will amend the expression.

Mark by the way just clarify on this, I am making my remarks based on my other post on convergence...and I had used the understanding of limits to conclude convergence...but was informed its not right approach.
Can one make use of limits in finding convergence of a series? I thought limits are applicable to sequences only...cheers

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