The root is around n+2/3

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