If sum of the series (wherever defined) is equal to

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

If sum of the series 1+xlog⁡∣1−sin⁡xcos⁡x∣(1+sin⁡xcos⁡x)1/2+x2log⁡∣1−sin⁡xcos⁡x∣(1+sin⁡xcos⁡x)1/4+…∞1 + x \log \left| \frac{1 - \sin x}{\cos x} \right| \left( \frac{1 + \sin x}{\cos x} \right)^{1/2} + x^2 \log \left| \frac{1 - \sin x}{\cos x} \right| \left( \frac{1 + \sin x}{\cos x} \right)^{1/4} + \dots \infty(wherever defined) is equal to k(1−x)(2−x)\frac{k(1-x)}{(2-x)} , then kk is equal to

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