<s>[INST] Why is $\lim_{n\to\infty} n(e - (1+\frac{1}{n})^n) = \frac{e}{2}$ I'm having trouble understanding why
$$\lim_{n\to\infty} n(e - (1+\tfrac{1}{n})^n) = \frac{e}{2}$$
Can someone offer me a proof for this?
 [/INST] Using: $\ln(1 + \frac{1}{n}) \sim \frac{1}{n} - \frac{1}{2n^2}$
And: $e^{-\frac{1}{2n}} \sim 1 - \frac{1}{2n} $
$$n(e - (1 + \frac{1}{n})^n) = n(e - e^{n\ln(1 + \frac{1}{n})}) = ne(1 - e^{n\ln(1 + \frac{1}{n}) - 1}) \sim ne(1 - e^{1 - \frac{1}{2n} - 1}) = ne(1 - e^{-\frac{1}{2n}}) = ne(1 - (1 - \frac{1}{2n}) = ne(\frac{1}{2n}) = \frac{e}{2}$$
$\therefore$ etc.
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