On the partial sums of a fourier series
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We give sharp lower estimates for the partial sums of the Fourier series sin x + cos2x/2 + sin 3x/3 + cos 4x/4 + ..., with both an even and odd number of terms. Our results are obtained through a monotonicity property of their local minima. In particular, we prove that all partial sums are positive on (0, π). © 2007 Springer.