Final answer:
The student's question involves finding the Fourier series of a function on a specific interval and determining its value at points of discontinuity. The process includes calculating the Fourier coefficients through integration and employing the Dirichlet conditions for convergence at discontinuities.
Step-by-step explanation:
The question is asking for the Fourier series of a function defined on a specific interval. The function appears to be piece-wise and may have discontinuities, which is important because the Fourier series converges to the midpoint of the jump at points of discontinuity per the Dirichlet conditions. To find the Fourier coefficients (a₀, aₙ, bₙ), we would integrate the function over the given interval, multiply by the relevant sine or cosine terms, and normalize by the interval's length. Then the series is constructed using these coefficients. If a point of discontinuity is present, Fourier series theory states that the series will converge to the average of the left-hand and right-hand limits of the function at that point.
When finding the location of the first focus by setting the distance to infinity, as suggested in the provided equation, we handle a situation typically associated with lens systems in optics, where Fourier optics is a relevant field; however, that is a separate context from the Fourier series of a function. Problems relating to dimensional consistency and series expansions, such as the binomial theorem, are also mathematical concepts but are again, not directly related to finding the Fourier series of a given function.