Prove, that sequence $ b_n = \sin1/2 +\sin2/4 +\dots+ \sin(n)/2^n$ is Cauchy

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I do not know, how to start proving that this sequence $$b_n = \sin1/2 + \sin2/4 +\dots+ \sin(n)/2^n$$ is Cauchy.

Thanks for all your answers.

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1 Answer

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Hint: if $n > m$,

$$|b_n - b_m| = \left|\frac{\sin (m+1)}{2^{m+1}}+\dots+\frac{\sin n}{2^n}\right| \leq \frac{1}{2^m}$$

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