How to prove that the limit of a sequence is a Wallis product?

In summary, the conversation is about finding the limit of a sequence, which is a Wallis product of 2/pi. The person asking the question is unsure of how to approach the problem, but eventually figures out the form of the Wallis product and uses it to solve the problem.
  • #1
nontradstuden
54
0

Homework Statement



Let t_1=1, t_(n+1)= [1- 1/4(n^2)]*t_n for n>=1.

The book says the limit is a Wallis product 2/pi, but I don't know where to start. I've been searching, but I'm lost. Could you point me in the right direction?
 
Last edited:
Physics news on Phys.org
  • #2
correction the limit is two over pi

my keyboard is messing up so i can not correct

I think I got it. I don't know if it's a proof, but I've found how it's the limit.
 
Last edited:
  • #3
Ok. I've found the form of a Wallis product...

lim n--> infinity (2n/ 2n-1) * (2n / 2n+1) = pi/2

so my sequence is pretty much the reciprocal of the equation that's why it's 2/pi... but I still don't know why it's 2/pi.

I saw some website saying this comes from some difficult integral, something about using method of reduction. I don't think I've ever done that in a calculus course before.

All help is appreciated! Thanks, folks.



I calculated up to t_7 to get

t_7= (143/144)*(99/100)*(63/64)*(35/36)*(15/…

so looking at the form of a Wallis product I changed it into

t_7= (11*13/ 12*12) * (9*11/10*10) * (7*9/8*8)* (5*7/6*6) * (3*5/4*4) *(1*3/2*2)

so that means t_n= (2n-1)*(2n+1)/ (2n)(2n), so the lim n-- infinity (t_n)= 2/pi.

Is this enough to consider this problem solved?
 

1. How do I define a Wallis product in a sequence?

A Wallis product is a specific type of infinite product that is used to calculate the limit of a sequence. It is defined as (2n/(2n-1)) * (2n/(2n+1)), where n is the index of the sequence. This product is used to approximate the value of pi and can be proven to converge to pi/2 as n approaches infinity.

2. What is the significance of proving the limit of a sequence as a Wallis product?

Proving the limit of a sequence as a Wallis product has significant implications in the field of mathematics. It is a fundamental tool in showing the convergence of a sequence and can be used to approximate the value of pi. It also has practical applications in fields such as physics, engineering, and economics.

3. How do I show that the limit of a sequence is a Wallis product?

To prove that the limit of a sequence is a Wallis product, you must first show that the sequence follows the pattern of a Wallis product. This can be done by substituting values for n and observing the trend of the sequence. Next, you must prove that the sequence converges to a specific value, in this case, pi/2. This can be done using mathematical techniques such as the squeeze theorem or the definition of a limit.

4. Can the Wallis product be used to calculate the value of pi?

Yes, the Wallis product can be used to approximate the value of pi. As n approaches infinity, the limit of the Wallis product will converge to pi/2. This value can then be multiplied by 2 to get an approximation of pi. The more terms in the product that are calculated, the closer the approximation will be to the actual value of pi.

5. Are there any limitations to using the Wallis product method to prove the limit of a sequence?

Yes, there are some limitations to using the Wallis product method. It can only be applied to certain types of sequences that follow the pattern of a Wallis product. Additionally, it can only be used to approximate the value of pi and cannot be used to calculate the exact value. Other methods, such as the Taylor series, may be more efficient for calculating the value of pi.

Similar threads

  • Calculus and Beyond Homework Help
Replies
5
Views
2K
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
850
  • Calculus and Beyond Homework Help
Replies
34
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
17
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
12
Views
2K
  • Calculus and Beyond Homework Help
Replies
11
Views
2K
Back
Top