If the sum of the series \( \sum_{n=1}^{\infty}\left(\frac{\pi-\sec...

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If the sum of the series \( \sum_{n=1}^{\infty}\left(\frac{\pi-\sec ^{-1} \sqrt{|x|+1}-\operatorname{cosec}^{-1} \sqrt{|x|+1}}{\pi a}\right)^{n} \) is finite, where
P
W \( x \in R \) and \( a0 \) then a lies in the interval :
(a) \( (1, \infty) \)
(b) \( \left(0, \frac{1}{2}\right) \)
(c) \( \left(\frac{1}{2}, \infty\right) \)
(d) \( \left[\frac{1}{2}, \infty\right) \)
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