<s>[INST] $\sin x-\frac{1}{2}\sin ^2x+\frac{1}{4}\sin ^3x-\frac{1}{8}\sin ^4x....\forall x \in \mathbb{R}$ is convergent? Is $\sin x-\frac{1}{2}\sin ^2x+\frac{1}{4}\sin ^3x-\frac{1}{8}\sin ^4x....\forall x \in  \mathbb{R}$ convergent? If it is convergent find the sum of the series.
Gives series $$\sin x-\frac{1}{2}\sin ^2x+\frac{1}{4}\sin ^3x-\frac{1}{8}\sin ^4x....=\sum _{n=1}^\infty \frac{(-1)^{n+1}}{2^{n-1}}\sin^n x$$
$$\left|\sum _{n=1}^\infty \frac{(-1)^{n+1}}{2^{n-1}}\sin^n x\right|\le \sum _{n=1}^\infty \frac{1}{2^{n-1}}=2$$ so is it convergent?
 [/INST] It converges to the limit $\frac{\sin x}{1+\frac{1}{2}\sin x}$ ;$a=\sin x$ and $r$(common ratio)$=-\frac{1}{2}\sin x.$ Since $|a|<1$.
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