The most general solution of  $2^{1+\left|\cos x\right|+{\left|\cos x\right|}^2+{\left|\cos x\right|}^3+\mathrm{...}\infty }=4is2n\pi \pm \frac{\pi }{a},thena=$

$We have 2^{1+\left|\cos x\right|+{\left|\cos x\right|}^2+{\left|\cos x\right|}^3+\mathrm{...}\infty }=4$

$1+\left|\cos x\right|+{\left|\cos x\right|}^2+.......\infty is in GP S_{\infty }=\frac{a}{1-r}$ 

                                                            $\begin{array}{l}\Rightarrow 2^{\frac{1}{1-\left|\cos x\right|}}=2^2\\ \Rightarrow \frac{1}{1-\left|\cos x\right|}=2\\ \Rightarrow \left|\cos x\right|=\frac{1}{2}\\ \Rightarrow \cos x=\pm \frac{1}{2}\Rightarrow x=2n\pi \pm \frac{\pi }{3},n\in z\end{array}$

                                                             $\Rightarrow a=3$