Errors are independent and normally distributed (for justification of least squares).
For ( n=1 ): coefficient ( 2 ) → matches sawtooth wave. ✔ At ( t=\pi/2 ): series gives ( 2 - 1 + 2/3 - 1/2 + \dots = \pi/2 ) (Leibniz series). ✔
Residuals: ( 2.1 - (1.233+1.35)= -0.483 ); ( 3.9 - (1.233+2.70)= -0.033 ); ( 5.8 - (1.233+4.05)= 0.517 ). Sum of residuals ≈ 0 (rounding). ✔
On average, ( y ) increases by 1.35 units per unit increase in ( x ), with an intercept of 1.233. Example 3 – Fourier Series (Periodic Forcing) Given: ( f(t) = t ) for ( -\pi < t < \pi ), extended periodically with period ( 2\pi ).
Errors are independent and normally distributed (for justification of least squares).
For ( n=1 ): coefficient ( 2 ) → matches sawtooth wave. ✔ At ( t=\pi/2 ): series gives ( 2 - 1 + 2/3 - 1/2 + \dots = \pi/2 ) (Leibniz series). ✔ fundamental applied maths solutions
Residuals: ( 2.1 - (1.233+1.35)= -0.483 ); ( 3.9 - (1.233+2.70)= -0.033 ); ( 5.8 - (1.233+4.05)= 0.517 ). Sum of residuals ≈ 0 (rounding). ✔ ✔ Residuals: ( 2
On average, ( y ) increases by 1.35 units per unit increase in ( x ), with an intercept of 1.233. Example 3 – Fourier Series (Periodic Forcing) Given: ( f(t) = t ) for ( -\pi < t < \pi ), extended periodically with period ( 2\pi ). Example 3 – Fourier Series (Periodic Forcing) Given:
