Final answer:
To obtain a sine series for the given function we extend it to be odd and periodically repeat this extension. The Fourier sine series approximates sin(z) and the coefficients decay generally according to a 1/n² rule.
Step-by-step explanation:
The function g(z) = 4/π² z (π - z) defined for 0 < z < π is not originally an odd function, nor does it have a period of 2π.
However, to construct a sine series that is odd and has a period of 2π, we extend g(z) to be odd about z = π and then periodically repeat this extension.
We derive the sine series coefficients for the odd extension of g(z) by integrating g(z) times sin(nz) over the interval from 0 to π, where n is a positive integer, and then multiplying by 2/π.
The result is an infinite sum of sine functions, known as a Fourier sine series.
The series begins to approximate sin(z) as more terms are taken.
However, g(z) does not exactly equal sin(z) for all z; hence, the approximation gets closer but is never perfect.
As n increases, the magnitude of the Fourier coefficients decays, generally following a 1/n² rule, due to the specific form of g(z) and its smoothness on the defined interval.