Proving Discrete Sum Equation - Step-by-Step Guide and Tips

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Homework Help Overview

The discussion revolves around proving a discrete sum equation, specifically involving trigonometric identities and properties of sums. The original poster seeks assistance in understanding a particular term in their proof related to the sum of cosine functions.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants explore the use of double angle formulas and Euler's formula to analyze the sum of cosine terms. There is a focus on understanding why a specific term in the expression is considered to be zero or negligible.

Discussion Status

The discussion is active with participants providing insights into the mathematical concepts involved. Some guidance has been offered regarding the application of Euler's formula and the nature of the sum of cosine terms, although there is no explicit consensus on the assumptions being made.

Contextual Notes

There are indications of differing interpretations regarding the significance of certain terms in the equation, particularly concerning their values and assumptions about their size.

darkfeffy
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Hi,

I need help in proving the equation in the attachment.

Thanks
darkfeffy
 

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Start by using a double angle formula to express sin(x)^2 in terms of cos(2x). Now get started.
 
Here is my work in the attachment. I just fail to see how the last term in the last expression (sum(cos(2pi*i/w))) is equal to 0.
 

Attachments

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    equation2.png
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Thinking it might just be an assumption that this last term is exceedingly small compared to the first.
 
darkfeffy said:
Thinking it might just be an assumption that this last term is exceedingly small compared to the first.

Do you know Euler's formula, e^(ix)=cos(x)+i*sin(x) (i the imaginary unit, not the integer index)? That would let you treat the sum of the cos term as the real part of the sum of a geometric series. And no, there's no approximation here. The cos part really does sum to zero.
 
Last edited:
darkfeffy said:
Thinking it might just be an assumption that this last term is exceedingly small compared to the first.
You are correct in that the term is exceedingly small (zero is an exceedingly small number). You are incorrect in that is an assumption.
 
Dick said:
Do you know Euler's formula, e^(ix)=cos(x)+i*sin(x) (i the imaginary unit, not the integer index)? That would let you treat the sum of the cos term as the real part of the sum of a geometric series. And no, there's no approximation here. The cos part really does sum to zero.
Thanks Dick.
 
D H said:
You are correct in that the term is exceedingly small (zero is an exceedingly small number). You are incorrect in that is an assumption.
Thanks DH for your brilliant reply which really doesn't add much information :-)
 

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