Summation for y(x)=Absolute value(sin(x)) -pi<x<pi

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The discussion focuses on deriving the Fourier series summation for the function y(x) = |sin(x)| within the interval -π < x < π. Participants clarify the relationship between cosine functions evaluated at specific multiples of π, specifically how cos(π*n) = (-1)^n and cos(π*(n-1)) also equals (-1)^(n+1). The interchangeability of these expressions is confirmed through algebraic manipulation, demonstrating that both yield the same result, thereby reinforcing the validity of the Fourier method in this context.

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i want to use the Fourier method to fins an summation for
y(x)=Absolute value(sin(x)) -pi<x<pi

i know that cos(pi*n)=(-1)^n

and get why they use cos(pi*(n+1))= (-1)^(n+1)

but why is then cos(pi*(n-1)) also (-1)^(n+1)


(see attachment)

THANX!
 

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look at the two expressions:

(-1)^(n+1) ==> (-1)^n * (-1)

and

(-1)^(n-1) ==> (-1)^n*(1/(-1)) = (-1)^n * (-1)

so they are interchangeable.
 
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