Problem 3. In this problem, we will determine a closed formula for the the n-th partial sum of a Fourier series of a function f : (-π, π) → R. You will need the following trigonometric identity (which you do not have to prove): 1 +cos u+cos 2u + ...cos nu = sin(n + 1)u 2 sin(u/2) 2 Suppose that f has period 27, with the Fourier series ao f(x) ~ 2 ∞ +Σ (ak cos(kx) + bk sin(kx)). k=1 If sn(x) denotes the n-th partial sum of this series, show that n 1 ao Sn(x) : 2 + Σ (a^ cos(kx) + by sin(kx)) ==—=—= √ f(t) sin[(n + 1)(t − x)] 2 sin((t - x)) dt. k=1

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