# Problematic Product Series!

Let $$P = \ \large{\prod_{k = 2}^{k \rightarrow \infty}(1 + \frac{(-1)^{k+1}(k-1)!}{a_k}})$$

where the sequence $$a_t$$ is given by the interesting recurrence relation $$a_2 = 2$$ and $$a_{t+1} = (t+1)(a_t + (-1)^{t+1}(t-1)!)$$ for all $$t \ge 2$$.

What's the value of $$\large e^P$$ where $$e$$ is the base of the natural logarithm?

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