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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