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à¤Ã×èͧÁ×ͧ͢ËÑÇ¢éÍ | ¤é¹ËÒã¹ËÑÇ¢é͹Õé |
#1
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[͹ءÃÁ] Åͧ·Ó¡Ñ¹´Ù
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#2
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ãªéÊٵâͧàÅͨͧ¡ìä´é¤ÃѺ
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#3
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ÁÕÍÕ¡¢éͨéÒÒ
$$ËÒ¤èҢͧ \lim_{n \to \infty} \frac{3n^3}{2}[\sum_{x = 1}^{n}(\sum_{y = 1}^{n}(\frac{(x+1)(x+y-1)!(n+1)!}{(n+x)!(y-1)!}))]^{-1}$$ |
#4
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¼ÁÅͧ´Ù·Õè summation µÑÇ㹡è͹¹Ð¤ÃѺ
´Ö§µÑÇ·ÕèäÁèÁÕ $y$ ÍÍ¡ÁÒ¡è͹àÅ ¨Ðä´é $\displaystyle A = \frac{(x+1)(n+1)!}{(n+x)!}\sum_{y = 1}^{n}\frac{(x+y-1)!}{(y-1)!}$ ¶éÒÁͧãËé $a=y-1$ ÍѴŧä»ã¹ summation ¨Ðä´é $\displaystyle \sum_{a = 0}^{n-1}\frac{(x+a)!}{(a)!}$ $\displaystyle= (x!)\sum_{a = 0}^{n-1}\frac{(x+a)!}{(a!)(x!)}$ $\displaystyle= (x!)\sum_{a = 0}^{n-1}\binom{x+a}{x}$ $\displaystyle= (x!)(\binom{x+n}{x+1})$ ¨Ðä´é $A= \frac{(x+1)(n+1)!(x!)}{(n+x)!}\times \frac{(x+n)!}{(x+1)!(n-1)!}$ $= n(n+1)$ àÍÒä»à¢éÒ summation µÑǵèÍä» «Ö觨Ðä´éÇèÒ¤èҢͧÁѹà»ç¹ $ n\times n(n+1) $ à¹×èͧ¨Ò¡äÁèÁÕ $x$ à»ç¹µÑÇÇÔè§ÍÕ¡ààÅéÇ $n(n+1)$ à»ÃÕºàËÁ×͹¤èÒ¤§·Õè ¨Ðä´éÇèÒ $\lim_{n \to \infty} \frac{3n^3}{2} \times \frac{1}{n(n)(n+1)}$ $= \frac{3}{2}$
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µéͧÊÙé¶Ö§¨Ðª¹Ð CCC Mathematic Fighting à¤ÃÕ´ àÅ |
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