Show Limit of Function: Find $\lim_{n \to \infty} = 1$

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Homework Statement


how to show \lim_{n \to \infty}\frac{\sum_{v=0}^{k}(-1)^v{k \choose v}e^{\sqrt{n-v}}}{2^{-k}n^{-\frac{1}{2}k}e^{\sqrt{n}}}=1, where n \geq k


Homework Equations


NIL


The Attempt at a Solution


I have absolutely no idea how to start.
 
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I'm guessing you have some typos up there. E.g. try it with k=1 - the limit isn't going to be 1.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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