Proving the Truth of a Mysterious Identity

  • Thread starter quasar987
  • Start date
  • Tags
    Identity
In summary, the conversation discusses a possible identity involving cosines and a geometric series. However, it is not confirmed to be true and there are some variations of the identity that are similar. One person suggests using a geometric expansion to find a closed form, while another mentions a thread where the correct form was discussed.
  • #1
quasar987
Science Advisor
Homework Helper
Gold Member
4,807
32
Someone told my friend, who in turn told me that this identity was true. However, I can't prove it, and when I try to use it I can't get the right answer to a rather simple problem. So, is it true that

[tex]\frac{1}{2} + \sum_{j=1}^n cos(jx) = \frac{sin([n+\frac{1}{2}]x)}{sin(\frac{x}{2})}[/tex]

?? Thx!
 
Mathematics news on Phys.org
  • #2
No, try n = x = 1.
 
  • #3
I can't tell you off-hand whether or not that identity is true, but at least there is a very similar identity (which might be equal to the one you posted). First, note that

[tex]\cos{jx} = Re(\cos{jx} + i \sin{jx}) = Re(e^{ixj}) = Re(({e^{ix}})^j).[/tex]

The Re function is linear, which means that summing cos(jx) is equivalent to summing (e^(ix))^j and then calculating the real part of that. Hence the problem can be reduced to calculating the sum of a geometric series...
 
Last edited:
  • #4
Pretty nifty identity, but you forgot a factor of 2. Since the left looks like a Fourier series, you can probably prove it by multiplying by cos(mx) and integrating
 
  • #5
It looks familiar. If I remember correctly, the name is the Dirichletkernel.

[tex]D_n:=\sum_{k=-n}^n e^{ikx}=1+2\sum_{k=1}^n \cos kx[/tex]

Use a geometric expansion to find a closed form. It only works if [itex]e^{ix}\not= 1[/itex]

[tex]D_n(x)=e^{-inx}\sum_{k=0}^{2n}e^{ikx}=\frac{\sin (n+1/2)x}{\sin x/2}[/tex]

If [itex]e^{ix}\not=1[/itex] you can expand the sum geometrically. After some algeblah you'll get the answer. Treat the case [itex]e^{ix}=1[/itex] seperately.
 
Last edited:
  • #7
K, thanx, I proved it by induction like the OP.
 

FAQ: Proving the Truth of a Mysterious Identity

What is an identity in science?

An identity in science is a statement or equation that is always true, regardless of the values or conditions involved. It is a fundamental concept in many scientific fields, including mathematics, physics, and chemistry.

How do scientists determine if an identity is true?

Scientists use a combination of experimentation, observation, and mathematical reasoning to determine if an identity is true. They test the identity under different conditions and compare the results to the expected outcome based on the identity. If the identity holds true in all cases, it is considered to be true.

Can an identity ever be proven wrong?

No, an identity cannot be proven wrong. It is a fundamental truth that is accepted based on evidence and logical reasoning. If evidence is found that contradicts an identity, it is either considered to be a flawed identity or it may prompt scientists to revise their understanding of the concept.

Are identities important in scientific research?

Yes, identities are crucial in scientific research as they serve as the foundation for many theories and laws. They provide a framework for understanding and predicting natural phenomena, and can also be used to develop new technologies and methods.

How do identities differ from hypotheses and theories?

An identity is a statement that is considered to be always true, while a hypothesis is a tentative explanation for a phenomenon that requires testing and evidence to support it. A theory, on the other hand, is a well-established explanation for a broad range of phenomena that has been extensively tested and supported by evidence.

Similar threads

Back
Top