Proving Convergence of Sequences at Infinity: A Case Study in Advanced Calculus

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In summary, the conversation discusses the convergence of three sequences (a, b, and c) at infinity. The sequence in part b converges to 1/2, and the goal is to find a way to solve for part c by finding another sequence that converges to a non-zero real number. The conversation also mentions using basic proving techniques and avoiding the use of derivatives or ratio tests. The speaker has already solved for part a and is trying to show that the sequence in part b is monotonically increasing and bounded by 1/2. They mention trying to directly show that a(n+1)-a(n) is always positive, but are struggling to do so. A suggestion is made to multiply the sequences by (sqrt
  • #1
WHOAguitarninja
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Homework Statement



Discuss the convergence of the following sequences at infinity.
a)[tex]\sqrt{n+1}[/tex]-[tex]\sqrt{n}[/tex]
b)[tex]\sqrt{n}[/tex]([tex]\sqrt{n+1}[/tex]-[tex]\sqrt{n}[/tex])
c)n([tex]\sqrt{n+1}[/tex]-[tex]\sqrt{n}[/tex])I've already solved a, and if I can solve b then I have c automatically.

This is for an undergraduate advanced calc course so all we've really covered is basic proving techniques, so I can't use derivatives or any sort of route/ratio test type thing.

I know the series converges to 1/2, so what I've done so far has been mostly aimed at trying to find another sequence that converges to a non 0 real number such that when I multiply or divide the above sequence (b) by it I get something easier to work with. This has proved unsuccesful. From examination and playing around it seems the sequence is monotonicly increasing, though I'm not entirely sure how to prove that without taking derivatives. I tried to directly show a(n+1)-a(n) is always positive where a(n) is the nth entry in the sequence. Still grinding that out but it's not going well, I don't think I'll get anything concrete.

Like I said...b is the key. If I can get b I can get c.

EDIT - Ok I just figured out a way to show it's monotonic. Now I just need to show it's bounded and it's lub is 1/2. Which is what I've been stuck on all day.
 
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  • #2
Multiply your sequences by (sqrt(n+1)+sqrt(n))/(sqrt(n+1)+sqrt(n)) and do some algebra.
 
  • #3
Dick said:
Multiply your sequences by (sqrt(n+1)+sqrt(n))/(sqrt(n+1)+sqrt(n)) and do some algebra.


You know...it's funny...I had considered multiplying by sqrt(n+1)+sqrt)n), but decided against it since it limits to infinity and I figured it wouldn't tell me anything...didn't even consider just multiplying what you suggested.

Thanks a million.
 

FAQ: Proving Convergence of Sequences at Infinity: A Case Study in Advanced Calculus

What is "convergence" in science?

Convergence in science refers to the process of multiple lines of evidence or data points coming together to support a single conclusion or theory. It is an important concept in scientific research as it provides a higher level of confidence in the validity of a hypothesis or theory.

Why is it sometimes difficult to prove convergence?

Proving convergence can be difficult because it requires a large amount of evidence from different sources to support a single conclusion. Additionally, there may be conflicting data or theories that make it challenging to determine the most accurate explanation for a phenomenon.

What are some common methods used to prove convergence?

Some common methods used to prove convergence include statistical analysis, experimental studies, and meta-analysis. These methods involve collecting and analyzing data from multiple sources to support a single conclusion.

Can convergence ever be proven definitively?

While convergence can be strongly supported by evidence, it is difficult to definitively prove due to the constantly evolving nature of science. New evidence or theories may emerge that challenge previously accepted conclusions, leading to a need for further research and refinement.

How does the concept of convergence relate to the scientific method?

The concept of convergence is closely tied to the scientific method as it involves gathering and analyzing evidence to support a hypothesis or theory. It is an essential part of the process of scientific inquiry and helps to increase the reliability and accuracy of scientific findings.

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