The root is around n+2/3

The root is around n+2/3

For every integer number $n$, denote by $c_n$ the (only one) positive root of the equation $\frac{e^x}{2}=\sum_{d=0}^n \frac{x^d}{d!}$. Prove that $\lim_{n\to \infty} c_n-n=\frac{2}{3}$

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