Relating products and taylor series

Calculus Level 5

\[ \prod_{m=1}^\infty \left [ \dfrac12 \exp \left(\sum_{n=1}^m \dfrac{ (-1)^{n-1}}n \right) \right ] = \dfrac{\exp(1)}2 \times \dfrac{\exp\left (1- \frac12\right)}2 \times\dfrac{\exp\left (1- \frac12+ \frac13\right)}2 \times\dfrac{\exp\left (1- \frac12+ \frac13- \frac14\right)}2 \times \cdots \]

Evaluate the infinite product above, where \(\exp(x) = e^x\).

If your answer can be expressed as \(r \times e^{s},\) where \(r\) and \(s\) are rational numbers, give your answer as \(r+s\).


Bonus: Give a closed form of \(\displaystyle \prod_{m=1}^\infty \left [ \dfrac1{1-x} \exp \left(-\sum_{n=1}^m \dfrac{x^{n}}n \right) \right ] \) for \( -1\leq x < 1 \).

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