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ILL 1990 Problem 35

    Prove that if ∣x∣<1|x| < 1 , then x(1−x)2+x2(1+x2)2+x3(1−x3)2+⋯=x1−x+2x21+x2+3x31−x3+⋯\frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots

    Dataset text AI-MO AoPS. Quoted with source.

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