The sequence $S_0,S_1,S_2$.... forms a G.P with common ratio

Given $y=e^{-x} \sin x$

$\begin{array}{l} -1 \leq \sin x \leq 1 \\ -e^{-x} \leq e^{-x} \sin x \leq e^{-x} \end{array}$

$\begin{aligned} x n & =n \pi ; n \in z \\ |y n| & =\mid y \text { coordinate of } S_{n} \mid \\ S_{n} & =(-1)^{n} \int_{n \pi}^{(n+1) \pi} e^{-x} \sin x d x \\ & =\frac{(-1)^{n+1}}{2}\left[e^{-x}(-\sin x+\cos x)\right]_{n \pi}^{(n+1) \pi}=\frac{e^{-n \pi}}{2}\left(1+e^{\pi}\right) \\ & \Rightarrow \frac{S_{n+1}}{S_{n}}=e^{-\pi} \\ S_{0} & =\frac{1}{2}\left(1+e^{\pi}\right) \\ \sum_{n=0}^{\infty} S_{n} & =\frac{1}{2}\left(1+e^{\pi}\right) \\ n=0 & \frac{1-e^{-\pi}}{.} \end{aligned}$

$S_{0}, S_{1}, S_{2} \ldots$ form an infinite GP with common ratio $e^{-\pi}$