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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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    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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Your question makes very little sense, especially the part about the compact interval $[a,b]$ - what does that have to do with $n$ and $N$, which it seems are both going to infinity?

Anyway, $\lim_{n\to\infty}\lim_{N\to\infty}(1+n+N)^{1/n}$ doesn't exist (or, if you prefer, is infinite), while $\lim_{N\to\infty}\lim_{n\to\infty}(1+n+N)^{1/n}=1$ If neither of these is what you want, please edit your question to clarify.

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    Gerry: Relax, I suffered briefly from the same hallucination but @André's comment saved me... :-)2012-08-26