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Snoop76   1
N Yesterday at 6:59 AM by Snoop76
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If $a_n={\sqrt[n]{n!!}}$ $,$$ $ find :$\lim_{n \to \infty}   \sqrt{n}|a_{n+1}+a_{n-1}-2a_n|$.
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Snoop76
Nov 11, 2024
Snoop76
Yesterday at 6:59 AM
Nice limit
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Snoop76
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If $a_n={\sqrt[n]{n!!}}$ $,$$ $ find :$\lim_{n \to \infty}   \sqrt{n}|a_{n+1}+a_{n-1}-2a_n|$.
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Snoop76
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Hint:
$(2n-1)!! = \frac{(2n)!}{(2n)!!} = \sqrt{2}\left(\frac{2n}{e}\right)^{n} \left(1 + O(\frac1{n})\right)$

$(2n)!! = 2^n n! = \sqrt{2n\pi}\left(\frac{2n}{e}\right)^{n} \left(1 +O(\frac1{n})\right)$
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