# Determining Convergence or Divergence

• Jul 6th 2008, 01:23 PM
auslmar
Determining Convergence or Divergence
Hello,

I'm having difficulty determining the behavior of this series.

"Show that the series: $\displaystyle \sum_{n=1}^{\infty} \frac{5n+1}{8n^2-10n-52}$ converges or diverges by finding a p-series, $\displaystyle \sum_{n=1}^{\infty}\frac{c}{n^p}$ or geometric series, $\displaystyle \sum_{n=1}^{\infty}cr^n$ to use in the comparison test.

1. The first item should be either the word less or the word greater.
2. The second item should be an expression for the term bn of the series used in the comparison test.
3. The third item should be a natural number N so that the comparison inequality an < bn holds for all n > N or an > bn holds for all n > N
4. The fourth item should be the word converges or the word diverges to indicate whether you are saying that the given series Sum(an,n=1..infinity, i.e., http://mapleta.okstate.edu/classes/c...an_1_2_inf.gif, converges or diverges."
I compared $\displaystyle \frac{5n+1}{8n^2-10n-52}$ with $\displaystyle \frac{5}{8n}$

saying that

$\displaystyle \frac{5n+1}{8n^2-10n-52}$ > $\displaystyle \frac{5}{8n}$ for n > 4.

And since $\displaystyle \frac{5}{8n}$ is divergent by p-series, that

$\displaystyle \frac{5n+1}{8n^2-10n-52}$ was also divergent.

greater,(5/8)(1/n),4,diverges

which is incorrect. If anyone can point out where I've gone wrong, I would greatly appreciate it.
• Jul 6th 2008, 01:54 PM
CaptainBlack
Quote:

Originally Posted by auslmar
Hello,

I'm having difficulty determining the behavior of this series.

"Show that the series: $\displaystyle \sum_{n=1}^{\infty} \frac{5n+1}{8n^2-10n-52}$ converges or diverges by finding a p-series, $\displaystyle \sum_{n=1}^{\infty}\frac{c}{n^p}$ or geometric series, $\displaystyle \sum_{n=1}^{\infty}cr^n$ to use in the comparison test.

1. The first item should be either the word less or the word greater.
2. The second item should be an expression for the term bn of the series used in the comparison test.
3. The third item should be a natural number N so that the comparison inequality an < bn holds for all n > N or an > bn holds for all n > N
4. The fourth item should be the word converges or the word diverges to indicate whether you are saying that the given series Sum(an,n=1..infinity, i.e., http://mapleta.okstate.edu/classes/c...an_1_2_inf.gif, converges or diverges."
I compared $\displaystyle \frac{5n+1}{8n^2-10n-52}$ with $\displaystyle \frac{5}{8n}$

saying that

$\displaystyle \frac{5n+1}{8n^2-10n-52}$ > $\displaystyle \frac{5}{8n}$ for n > 4.

And since $\displaystyle \frac{5}{8n}$ is divergent by p-series, that

$\displaystyle \frac{5n+1}{8n^2-10n-52}$ was also divergent.

greater,(5/8)(1/n),4,diverges

which is incorrect. If anyone can point out where I've gone wrong, I would greatly appreciate it.

Well that looks OK to me, so perhaps its just being over restrictive in what it will accept, try:

greater,5/(8n),4,diverges

or:

greater,5/(8*n),4,diverges

or:

greater,(5/8)(1/n),3,diverges

(3 will do here since strict inequalities are involved, but the answer would be right for any value greater than or equal to 3)

RonL
• Jul 6th 2008, 01:59 PM
auslmar
Thanks for the help.

Actually, it accepted:

greater,(5)/(8*n),5,diverges
• Jul 6th 2008, 03:41 PM
Plato
Quote:

Originally Posted by auslmar
it accepted:
greater,(5)/(8*n),5,diverges

That is odd! That change is exactly what I had said in the post that I deleted.
But when I ran it through MathCad, N>=4 worked.
I wonder if the author of this problem made the same mistake that I did.
• Jul 6th 2008, 07:52 PM
CaptainBlack
Quote:

Originally Posted by Plato
That is odd! That change is exactly what I had said in the post that I deleted.
But when I ran it through MathCad, N>=4 worked.
I wonder if the author of this problem made the same mistake that I did.

There are two changes from the original submission I wonder if it needed both, or if just one of them would have worked?

RonL