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Prove that the infinite series 1\minus{}\frac{1}{x(x\plus{}1)}\minus{}\frac{x\minus{}1}{2!x^2(2x\plus{}1)}\minus{}\frac{(x\minus{}1)(2x\minus{}1)}{3!(x^3(3x\plus{}1))}\minus{}\frac{(x\minus{}1)(2x\minus{}1)(3x\minus{}1)}{4!x^4(4x\plus{}1)}\minus{}\cdots is convergent for every positive

x

. Denoting its sum by

F(x)

, find \lim_{x\to \plus{}0}F(x) and

\lim_{x\to \infty}F(x)

Problem Statement