Let f be a real-valued function such that f, f 0 , and f 00 are all continuous on [0, 1]. Consider the series P∞ k=1 f( 1 k ). (a) Prove that if the series P∞ k=1 f( 1 k ) is convergent, then f(0) = 0 and f 0 (0) = 0. (b) Conversely, show that if f(0) = f 0 (0) = 0, then the series P∞ k=1 f( 1 k ) is convergent.
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