Home Competitions SCHWEITZER 2000 Problem 5
SCHWEITZER 2000 Problem 5
Prove that for every ε > 0 \varepsilon >0 ε > 0 n n n a 1 , … , a n a_1, \ldots, a_n a 1 , … , a n ε < x < 2 π − ε \varepsilon < x < 2\pi - \varepsilon ε < x < 2 π − ε ∑ k = 1 n a k cos k x < − 1 ε ∣ ∑ k = 1 n a k sin k x ∣ \sum_{k=1}^n a_k\cos kx < -\frac{1}{\varepsilon}\left| \sum_{k=1}^n a_k\sin kx\right| k = 1 ∑ n a k cos k x < − ε 1 k = 1 ∑ n a k sin k x
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