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

[darkfeffy]

Hi,

I need help in proving the equation in the attachment.

Thanks
darkfeffy

 

[Dick]

Start by using a double angle formula to express sin(x)^2 in terms of cos(2x). Now get started.

 

[darkfeffy]

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.

 

[darkfeffy]

Thinking it might just be an assumption that this last term is exceedingly small compared to the first.

 

[Dick]

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.

 

[D H]

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.

 

[darkfeffy]

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.

 

[darkfeffy]

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 :-)