I have two sequences and I need to find their limits
$a_n = \sqrt[n]{e^n - (\sqrt{e})^n}$
$b_n = \sqrt[n]{e^n - (1 + \frac{1}{2n})^{n^2}}$
So I know that the limit is $e$, but I don't know how to show it
2026-04-12 01:59:07.1775959147
Show that $\lim_{n \to +\infty} \sqrt[n]{e^n - (\sqrt{e})^n} = e$
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Hint
$$a_n = \sqrt[n]{e^n - (\sqrt{e})^n}= \sqrt[n]{e^n(1-e^{-n/2})}=e\sqrt[n]{1-\frac{1}{e^{n/2}}}$$