Determine whether the following series converges

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I didn't forget it, i just made a mistake in the working. I'll try to be more careful in future. Thanks for the help.In summary, the given series diverges by the limit comparison test.
  • #1
elliotician
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Homework Statement



Determine whether the following series converges:

[tex]
\sum \frac{(n-1)^3}{\sqrt{n^8+n+2}}
[/tex]

Homework Equations



Definition of convergence:

Let [tex]\sum a_{n}[/tex] be a series.

If the sequence of (sn) partial sums converges to L (finite). Then we say the series converges to L or has sum L. If (sn) diverges we say [tex]\sum a_{n}[/tex] diverges.

The Attempt at a Solution



With some manipulation i can see the sequence acts like 1/n, thus the series [tex]\sum a_{n}[/tex] would diverge. However i can't use 1/n as a comparison since we would not have an > bn. So I haven't been able to find a suitable divergent sequence.

So basically we need a sequence bn such:

[tex]\frac{(n-1)^3}{\sqrt{n^8+n+2}}}>b_{n}[/tex]

This doesn't need to hold for all natural numbers n we can have it for some n>=a.

And bn must of course be divergent
 
Last edited:
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  • #2


Next question, if someone can tell me why my LaTex isn't working that would be fantastic ?
 
  • #3


Fixed your LaTeX. See below for what I did.
elliotician said:

Homework Statement



Determine whether the following series converges:

[tex]
\sum \frac{(n-1)^3}{\sqrt{n^8+n+2}}
[/tex]


Homework Equations



Definition of convergence:

Let [tex]\sum a_{n}[/tex] be a series.

If the sequence of (sn) partial sums converges to L (finite). Then we say the series converges to L or has sum L. If (sn) diverges we say [tex]\sum a_{n}[/tex] diverges.


The Attempt at a Solution



With some manipulation i can see the sequence acts like 1/n, thus the series [tex]\sum a_{n}[/tex] would diverge. However i can't use 1/n as a comparison since we would not have an < bn. So I haven't been able to find a suitable divergent sequence.

So basically if the is a sequence bn such:

[tex]\frac{(n-1)^3}{\sqrt{n^8+n+2}}}<b_{n}[/tex]

This doesn't need to hold for all natural numbers n we can have it for some n>=a.

Some of your beginning tex tags had a '\' character in them, and all of you ending tex tags had \ instead of /.

Regarding your problem, do you know the Limit Comparison Test? If you can show that the arbitrary term in your series is > 1/n, the Limit Comparison Test might be helpful.
 
  • #4


Cheers, and crikey i made loads of mistakes in there - fixing :)
 
  • #5


I think the arbitrary term in my series is less than 1/n except for very small values of n below zero, so 1/n isn't suitable.

With some manipulation we can express the nth term as as:

[tex]
\frac{(1-1/n)^3}{n\sqrt{1+n^{-7}+2n^{-8}}}
[/tex]
 
Last edited:
  • #6


I'm not thinking clearly at the moment, need a break, but I think 1/4n would work though for n>=3.
 
  • #7


You mean 1/n isn't suitable for the Limit Comparison Test?
 
  • #8


elliotician said:
I think the arbitrary term in my series is less than 1/n except for very small values of n below zero, so 1/n isn't suitable.

With some manipulation we can express the nth term as as:

[tex]
\frac{(1-1/n)^3}{n\sqrt{1+n^{-7}+2n^{-8}}}
[/tex]
If the arbitrary term in your series is less than 1/n, then the Comparison Test is no help, but try the Limit Comparison Test, if you know it. There is also the Ratio Test, which might be useful.

Also, you probably aren't interested in negative values of n. The index for series usually starts at 0 or 1. Not always, but usually.
 
  • #9


Mark44 said:
If the arbitrary term in your series is less than 1/n, then the Comparison Test is no help

I do know the limit comparison however i just get the sequence tending to 0 rather than a finite L. So doesn't seem to be of use

Comparison test should work for 1/4n, for n>=2

[tex]\frac{(1-1/n)^{3}}{n\sqrt{1+n^{-7}+2n^{-8}}}\geq\frac{1}{4n}\][/tex]

So by comparison test the series diverges?
 
  • #10


Bohrok said:
You mean 1/n isn't suitable for the Limit Comparison Test?

No i meant for the comparison test, however if i use 1/n in the limit comparison i get an/bn tending to 0
 
  • #11


elliotician said:
I do know the limit comparison however i just get the sequence tending to 0 rather than a finite L. So doesn't seem to be of use

Comparison test should work for 1/4n, for n>=2

[tex]\frac{(1-1/n)^{3}}{n\sqrt{1+n^{-7}+2n^{-8}}}\geq\frac{1}{4n}\][/tex]

So by comparison test the series diverges?

You'd have to prove or show how you got that inequality in order to use it for the comparison test.

If you use 1/n for the limit comparison test, you should not get 0 if you start with your manipulated series.
 
  • #12


I don't see your work using the limit comparison test, so can't comment on the correctness of your result. You shouldn't get a sequence that tends to zero -- maybe you meant the limit was zero.

That test works this way, with an being the general term in your series and bn being the one in 1/n.

If [tex]\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = L \neq 0[/tex]
and L is a finite number, then both series converge, or they both diverge. The wikipedia article (http://en.wikipedia.org/wiki/Limit_comparison_test) is somewhat misleading about when you can use this test. It's not just for convergent series. The external reference at the end of the wikipedia article I cited has a more complete presentation.

I haven't calculated this limit, so I can't tell if this test would have been conclusive. If you can actually prove the inequality you have in your last post, the comparison test tells you that your series diverges.
 
  • #13


Ah no i made a mistake! (again!)

Limit comparison works, tends to 1 when i divide by 1/n.

Thanks!
 
  • #14


So to conclude

By the standard comparison test:

since (1-1/n)^3 >= 1/2 for n>=2

and [tex]n\sqrt{1+n^{-7}+2n^{-8}[/tex] <= 2n

we must have
[tex]
\frac{(1-1/n)^{3}}{n\sqrt{1+n^{-7}+2n^{-8}}}\geq\frac{1/2}{2n}=\frac{1}{4n}\]
[/tex]

More simply by the limit comparison test

[tex]\frac{(1-1/n)^{3}/n\sqrt{1+n^{-7}+2n^{-8}}}{1/n}\rightarrow\frac{1/n}{1/n}=1\][/tex]
 
  • #15


elliotician said:
So to conclude

since (1-1/n)^3 >= 1/2 for n>=2

That's not true, try it for n=2 and n=3. Are you forgetting the ^3?
 
  • #16


Bohrok said:
That's not true, try it for n=2 and n=3. Are you forgetting the ^3?

I meant n>=5 :)
 

1. What is a series?

A series is a mathematical concept that involves adding up an infinite number of terms. It is denoted by the symbol ∑ and can be written in the form ∑f(n) where f(n) is a function representing the terms of the series.

2. How do you determine if a series converges?

To determine if a series converges, you can use various tests such as the ratio test, the comparison test, or the integral test. These tests involve evaluating the behavior of the terms in the series to see if they approach a finite limit or if they are bounded by a convergent series.

3. What does it mean if a series converges?

If a series converges, it means that the sum of the infinite number of terms in the series approaches a finite limit. This implies that the series has a well-defined value and can be calculated accurately to a certain degree of precision.

4. What does it mean if a series diverges?

If a series diverges, it means that the sum of the infinite number of terms in the series does not approach a finite limit. This can happen if the terms in the series increase without bound or if they oscillate between positive and negative values without settling on a specific value.

5. Can a series converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series can only have one of these two outcomes - either it converges to a finite limit or it diverges to infinity. It is not possible for a series to have both behaviors simultaneously.

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