Wikipedia:Reference desk/Archives/Mathematics/2013 January 27

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January 27

Limit 2

$\lim _{n\rightarrow \infty }\sum _{k=0}^{n}{\frac {k^{k}(n-k)^{n-k}}{k!(n-k)!}}n!n^{-n-1/2}$

Already simplified as much as I can. --AnalysisAlgebra (talk) 17:50, 27 January 2013 (UTC)

In this context, $0^{0}:=1$ --AnalysisAlgebra (talk) 17:53, 27 January 2013 (UTC)

Have you tried using Stirling's approximation? Sławomir Biały (talk) 18:17, 27 January 2013 (UTC)

Yes, I have. That gives $\lim _{n\rightarrow \infty }\sum _{k=0}^{n}{\frac {k^{k}(n-k)^{n-k}}{k!(n-k)!}}{\sqrt {2\pi }}e^{-n}$. I don't see how that makes it any easier. --AnalysisAlgebra (talk) 18:20, 27 January 2013 (UTC)

Would it be allowed to apply Stirling for k and (n-k) as well? If you substitute k^k and (n-k)^(n-k), you get a much simpler formula it seems. But I don't know if that substitution is valid ... Ssscienccce (talk) 01:48, 28 January 2013 (UTC)

The approximation is only valid for large $k$ or $(n-k)$, but they are not all large (k is taken from 0). --AnalysisAlgebra (talk) 11:59, 28 January 2013 (UTC)

You can in fact use Stirling's formula here, for a simple reason - it's indeed inaccurate for small k and n-k, but as $n\to \infty$ the summands where k or n-k are small become a vanishing part of the entire sum, so their accuracy has no effect on the result for the limit. -- Meni Rosenfeld (talk) 14:08, 29 January 2013 (UTC)

Have you tried applying the binomial theorem? Sławomir Biały (talk) 20:02, 27 January 2013 (UTC)

How could you use that? --AnalysisAlgebra (talk) 03:25, 28 January 2013 (UTC)

Observation, if it is helpful...     ${\frac {k^{k}(n-k)^{n-k}}{k!(n-k)!}}n!n^{-n-1/2}={\frac {n!}{k!(n-k)!}}.{\frac {k^{k}(n-k)^{n}}{(n-k)^{k}}}.{\frac {1}{n^{n}{\sqrt {n}}}}={\frac {n!}{k!(n-k)!}}.{\frac {(1-{\frac {k}{n}})^{n}}{({\frac {n}{k}}-1)^{k}}}.{\frac {1}{\sqrt {n}}}$.     EdChem (talk) 02:59, 28 January 2013 (UTC)

It's not clear to me that the limit exists, but numerically it looks as though it does. Using R, the values for n=1e6,2e6 and 4e6 give 1.253981, 1.253786 and 1.253648 respectively. HTH, Robinh (talk) 08:20, 28 January 2013 (UTC)

continuous compounding formula derivation proof

Is this how its done or is there another simpler way?

link — Preceding unsigned comment added by Ap-uk (talk • contribs) 23:43, 27 January 2013 (UTC)

That is OK but not really rigorous.

$P(t)=\lim _{m\to \infty }P_{0}\left(1+{\frac {r}{m}}\right)^{mt}$
Substitute $n={\frac {m}{r}}$. No matter what r is as long as it's positive, as m→∞, n→∞, so you have
$P(t)=\lim _{n\to \infty }P_{0}\left(1+{\frac {1}{n}}\right)^{(rn)t}$
$P(t)=P_{0}\left(\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}\right)^{rt}$
We can define $e:=\lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}$
so $P(t)=P_{0}e^{rt}$
72.128.82.131 (talk) 02:12, 28 January 2013 (UTC)