Series for erf with exponential factored out

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Theorem: The following formula holds: $$\mathrm{erf}(z)=\dfrac{2}{\sqrt{\pi}}e^{-z^2}\displaystyle\sum_{k=0}^{\infty} \dfrac{2^k}{1 \cdot 3 \cdot \ldots \cdot (2k+1)} z^{2k+1},$$ where $\mathrm{erf}$ denotes the error function and $\pi$ denotes pi.

Proof: