Recurrent sequence
Recurrent sequence
Let \(a_n\) be defined by \(\displaystyle{a_n=n(n-1)a_{n-1}+\frac{n(n-1)^2}{2}a_{n-2}}\) for \(n\geq3\) and \(a_{1}=0,\;a_2=1\).
\(\displaystyle{1)}\) Show that \(\displaystyle{\lim_{n\to+\infty}\frac{e^{2n}a_n}{n^{2n+1/2}}=2\sqrt{\frac{\pi}{e}}}\) and
\(\displaystyle{2)}\) compute \(\displaystyle{\lim_{n\to+\infty}n\left(\frac{e^{2n}a_n}{n^{2n+1/2}}-2\sqrt{\frac{\pi}{e}}\right)}\) if it exists.
\(\displaystyle{1)}\) Show that \(\displaystyle{\lim_{n\to+\infty}\frac{e^{2n}a_n}{n^{2n+1/2}}=2\sqrt{\frac{\pi}{e}}}\) and
\(\displaystyle{2)}\) compute \(\displaystyle{\lim_{n\to+\infty}n\left(\frac{e^{2n}a_n}{n^{2n+1/2}}-2\sqrt{\frac{\pi}{e}}\right)}\) if it exists.
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