Let f(x) = a1 sin x + a2 sin 2x + . . . + an sin nx, where a1, a2, . . ., an are real numbers and where n is a positive integer. Given that ∣f(x)∣ ≤ ∣sin x∣ for all real x, prove that ∣a1 + 2a2 + . . . + nan∣ ≤ 1.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.3: Factorization In F [x]
Problem 24E
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Let f(x) = a1 sin x + a2 sin 2x + . . . + an sin nx, where a1, a2, . . ., an are real numbers and where n is a positive integer. Given that ∣f(x)∣ ≤ ∣sin x∣ for all real x, prove that ∣a1 + 2a2 + . . . + nan∣ ≤ 1. 

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