Is this Infinite Product True or Not?

  • Context: Graduate 
  • Thread starter Thread starter Werg22
  • Start date Start date
  • Tags Tags
    Product
Click For Summary

Discussion Overview

The discussion revolves around the validity of an infinite product representation of the sine function, specifically whether the proposed forms accurately represent sin(x) and converge appropriately. The scope includes mathematical reasoning and exploration of infinite products.

Discussion Character

  • Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant proposes an infinite product representation of sin(x) as f(x)=sin(x)=x(x-pi)(x-2pi)(x-3pi)(x-4pi)...
  • Another participant questions the validity of this representation by suggesting testing specific values like x = 2pi and x = pi.
  • A later reply indicates that the proposed infinite product does not converge and provides an alternative infinite product form for sine: \sin(x)=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{(\pi n)^2} \right).
  • Some participants express confusion regarding the implications of the infinite series and its behavior at specific points, particularly noting that sin(pi) = 0, which raises concerns about the convergence of the proposed series.
  • Repeated clarifications from the original poster indicate a misunderstanding in their initial formulation of the infinite product.
  • Another participant reiterates that the proposed series does not converge, stating that the absolute value of the terms is growing without bound.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the validity of the infinite product representation, with multiple competing views regarding its convergence and correctness remaining unresolved.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about convergence and the definitions of the infinite product forms. The implications of specific values on the proposed series are also not fully explored.

Werg22
Messages
1,431
Reaction score
1
Is this true?

f(x)=sin(x)=x(x-2pi)(x-4pi)(x-6pi)...

edit:

f(x)=sin(x)=x(x-pi)(x-2pi)(x-3pi)(x-4pi)...
 
Last edited:
Mathematics news on Phys.org
It doesn't seem right to me, try x = 2pi.
 
Sorry I did a mistake. I meant sin(x).
 
Try x = pi.
 
I'm affraid I do not see what you mean... the serie is infinite...
 
No, your infintite product does not even converge.

Infinite product form for sine:

[tex]\sin(x)=x\prod_{n=1}^{\infty}\left(1-\frac{x^2}{(\pi n)^2} \right)[/tex]
 
Last edited:
Werg22 said:
I'm affraid I do not see what you mean... the serie is infinite...
I was trying to point out that sin(pi) = 0 but your serie would never go to 0.
 
Sorry what I really meant is f(x)=sin(x)=x(x-pi)(x-2pi)(x-3pi)(x-4pi)...

Really sorry.
 
Werg22 said:
Sorry what I really meant is f(x)=sin(x)=x(x-pi)(x-2pi)(x-3pi)(x-4pi)...

Really sorry.

Still doesn't converge, the absolute value of the terms is growing without bound.
 

Similar threads

High School Curiosity on this infinite product
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Why are 0 and pi/2 the solutions for sin(x)=0 and cos(x)=0?
  • · Replies 4 ·
Replies
4
Views
2K
High School Notation for infinite iteration
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Alternating Harmonic Numbers are cool, spread the word!
  • · Replies 4 ·
Replies
4
Views
4K
MHB Interpolating Points with Continuous Modular Functions?
  • · Replies 5 ·
Replies
5
Views
2K
MHB What are the domain and range for the inverse function $f^{-1}$ of $f$?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Product of functions to express any function
  • · Replies 5 ·
Replies
5
Views
2K
MHB Evaluate (without a calculator) the product: f(r1+1)⋅f(r2+1)⋅f(r3+1).
  • · Replies 10 ·
Replies
10
Views
3K
MHB Proving an Infinite Number of Integer Points on a Level Surface
  • · Replies 1 ·
Replies
1
Views
2K
MHB Prove The Product Is Greater Than 5
  • · Replies 4 ·
Replies
4
Views
1K