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Consider the (2\pi)-periodic function (f(t)=t^2) defined on ([-\pi,\pi]). Since (t^2) is an even function, its Fourier series contains only cosine terms. Thus the Fourier series has the form [ f(t)\sim \frac{a_0}{2}+\sum_{n=1}^{\infty} a_n\cos(nt), ] where [ a_0=\frac{1}{\pi}\int_{-\pi}^{\pi} t^2,dt,\qquad a_n=\frac{1}{\pi}\int_{-\pi}^{\pi} _______ __________ _________ ____ ___ _______.
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