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It is known that for integers $n \geq 1$,

$$\lim_{ n \to \infty} (1 + n)^{1/n} = e = 2.718\dots$$

For integer $N \ge n$, is it true that:

$$\lim_{ n, N \to \infty} (1 + n + N)^{1/n} > e\ \ ?$$

Suppose the sequence is monotone either increasing or decreasing and also that infinitely many terms of the limit

$$\lim_{ n, N \to \infty} (1 + n + N)^{1/n}$$

are bounded within some compact interval $[a, b]$ on the real line. Is this limit

$$\lim_{ n, N \to \infty }(1 + n + N)^{1/n}$$

finite on $[a, b]$? Does it converge to some finite value?

If anyone can help to solve this question then I thank you in advance.

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    The answers depend on whether you take the limits with respect to $n$ or $N$ first. (If you do both at the same time the limit won't exist, because it would require the two differently-ordered limits to be equivalent at least.)2012-08-25
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    The limit you gave has value $1$, not $e$. You want $(1+1/n)^n$.2012-08-25
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    I think he really wants $(1+n)^{1/n}$, but with $n\rightarrow 0$. Maybe not...i dont know.2012-08-25
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    You originally had $n=\gt1$ and $N=\gt n$; different people changed this to $n\ge1$ and $N\to n$, respectively; it seems unlikely that this was what you intended. I've now changed $N\to n$ to $N\ge n$; please check whether this is what you wanted. To avoid all this confusion in the future, please use $\TeX$ to format your posts yourself. Inline formulas are enclosed in single dollar signs, displayed equations in double dollar signs; you can get the code for any math formatting you see on this site by right-clicking on it and selecting "Show Math As:TeX Commands".2012-08-26

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