Determining if the sequence convergers or diverges(II)

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In summary, convergence in a sequence means the terms approach a specific limit while divergence means they do not. To determine convergence or divergence, methods like the limit comparison test can be used. A sequence cannot both converge and diverge, but it can be indeterminate. Absolute convergence occurs when the absolute values converge, while conditional convergence involves the signs of the terms. Real-life applications of determining convergence or divergence can be found in finance, physics, and engineering, such as in compound interest, describing motion, and analyzing stability of structures.
  • #1
shamieh
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I'm confused on how they are getting their result...

Determine if the sequence converges or diverges, if it converges, find the limit...

\(\displaystyle \frac{n^2}{2n - 1} - \frac{n^2}{2n + 1}\)

So I started plugging in from 1 because it looks like they want me to do something with a telescoping series and I got:

(1 - 1/3) + (4/3 - 4/5) + (9/5 - 9/7) + ... (Which really got me no where?)

Then I tried to just take the limit as n -> infinity using l'opitals and ended up with \(\displaystyle n - n\) which got me no where as well...So I'm really confused on how they know that it converges to \(\displaystyle \frac{1}{2}\)
 
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  • #2
shamieh said:
I'm confused on how they are getting their result...

Determine if the sequence converges or diverges, if it converges, find the limit...

\(\displaystyle \frac{n^2}{2n - 1} - \frac{n^2}{2n + 1}\)

So I started plugging in from 1 because it looks like they want me to do something with a telescoping series and I got:

(1 - 1/3) + (4/3 - 4/5) + (9/5 - 9/7) + ... (Which really got me no where?)

Then I tried to just take the limit as n -> infinity using l'opitals and ended up with \(\displaystyle n - n\) which got me no where as well...So I'm really confused on how they know that it converges to \(\displaystyle \frac{1}{2}\)

What you've been given is NOT a series, so why would you try to see if this non-existent series is telescopic?

The right thing to do with this sequence is to take the limit as n approaches infinity.

$\displaystyle \begin{align*} \frac{n^2}{2n -1} - \frac{n^2}{2n + 1} &= \frac{n^2 \left( 2n + 1 \right) - n^2 \left( 2n - 1 \right) }{ \left( 2n -1 \right) \left( 2n + 1 \right) } \\ &= \frac{2n^3 + n^2 - 2n^3 + n^2}{4n^2 - 1} \\ &= \frac{2n^2}{4n^2 - 1} \\ &= \frac{2}{4 - \frac{1}{n^2} } \\ &\to \frac{2}{4 - 0} \textrm{ as } n \to \infty \\ &= \frac{1}{2} \end{align*}$

So the sequence converges to 1/2.
 
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  • #3
Prove It said:
What you've been given is NOT a series, so why would you try to see if this non-existent series is telescopic?So the series converges to 1/2.

Yes you are correct I am an idiot. Idk why I tried that. I'm assuming you mean the sequence converges though right? Just a typo? Again, thank you for the explanation. Helped me so much!
 
  • #4
Yes that was a typo, I fixed it now :)
 

FAQ: Determining if the sequence convergers or diverges(II)

1. What is the definition of convergence and divergence in a sequence?

Convergence in a sequence means that the terms of the sequence approach a specific limit as the number of terms increases, while divergence means that the terms of the sequence do not approach a specific limit and instead continue to increase or decrease without bound.

2. How do you determine if a sequence converges or diverges?

To determine if a sequence converges or diverges, you can use various methods such as the limit comparison test, ratio test, or root test. These methods involve evaluating the limit of the sequence and comparing it to known convergent or divergent sequences.

3. Can a sequence converge and diverge at the same time?

No, a sequence can either converge or diverge but not both. If a sequence converges, it cannot diverge and vice versa. However, it is possible for a sequence to neither converge nor diverge, in which case it is considered indeterminate.

4. What is the difference between absolute and conditional convergence?

A sequence is absolutely convergent if the absolute value of its terms converges, while it is conditionally convergent if the sequence itself converges but not its absolute value. In other words, if a sequence is conditionally convergent, the signs of its terms play a role in its convergence or divergence.

5. Are there any real-life applications of determining if a sequence converges or diverges?

Yes, determining if a sequence converges or diverges is important in many fields such as finance, physics, and engineering. For example, in finance, the concept of compound interest relies on the convergence of a sequence. In physics, the motion of an object can be described using convergent and divergent sequences. In engineering, the stability of structures can be determined by analyzing the convergence or divergence of a sequence.

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