As you know, is an irrational number that can only be approximated by a number with

As you know, π is an irrational number that can only be approximated by a number with a finite number of decimals. How to compute this value recursively is a problem of theoretical interest. In this problem we show that the Fourier series can provide that formulation.

(a) Consider a train of rectangular pulses x(t), with a period

x₁(t) = 2[u(t + 0.25) − u(t − 0.25)] − 1,                      −0.5 ≤ t ≤ 0.5

and period T₀ = 1. Plot the periodic signal and find its trigonometric Fourier series.

(b)Use the above Fourier series to find an infinite sum for π.

(c) If π_N is an approximation of the infinite sum with N coefficients, and πis the value given by , find the value of N so that π_N is 95% of the value of π given by MATLAB.