I dont know how to evaluate the first one, for second one I can only show the sum is less than 2. $$\begin{align} & \prod\limits_{k=4}^{\infty }{\left( 1-{{\left( \frac{3}{k} \right)}^{3}} \right)} \\ & \sum\limits_{n=1}^{\infty }{\sin \left( \frac{1}{2^{n-1}} \right)} \\ \end{align}$$
2026-05-14 10:07:05.1778753225
Sum of series & sequences
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Here is a result by maple for the infinite product
$$\prod\limits_{k=4}^{\infty }{\left( 1-{{\left( \frac{3}{k} \right)}^{3}} \right)} = {\frac {8}{15561\,\pi}}\,{ {\cosh \left( \frac{3\pi \,\sqrt {3}}{2} \right) }}. $$
Added: The numerical sum of the series evaluated by maple
$$ \sum\limits_{n=1}^{\infty }{\sin \left( \frac{1}{2^{n-1}} \right)} \sim 1.817928721 $$