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⨷Âì¹Õé à¡Ô´¨Ò¡¡ÒäԴàÅè¹æ¢Í§¼Áàͧ «Öè§â¨·ÂìÁÕÍÂÙèÇèÒ (µÑǼÁàͧÂѧäÁèÊÒÁÒöËҤӵͺä´é ÂѧËÒÍÂÙè)
¡Ó˹´ãËé $A_1$ à»ç¹Í¹Ø¡ÃÁ͹ѹµì «Öè§ËÒ¤èÒä´é áÅÐ $\frac{2}{3}A_n = \frac{1}{3^{2n-1}}+A_{n+1}$ ·Ø¡¨Ó¹Ç¹àµçÁ $n \geqslant 1$ ¶ÒÁÇèÒ ÊÒÁÒö ËÒ¤èÒ $\lim_{n \to \infty} A_n$ áÅФèÒ·Õèá¹è¹Í¹¢Í§ $A_1$ ä´éËÃ×ÍäÁè áÅФèÒ¹Ñé¹æ ÁÕ¤èÒà·èҡѺà·èÒäà à¡Ô´¨Ò¡¡ÒäԴàÅè¹æ¢Í§¼Áàͧ ¶éÒ⨷Âì¼Ô´¾ÅÒ´»ÃСÒÃã´ ¢Íâ·É´éǤÃѺ T T
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àÁ×èÍäÃàÃÒ¨Ðà¡è§àÅ¢¹éÒÒÒÒÒÒ ~~~~ T T äÁèà¡è§«Ñ¡·Õ ·Óä§´Õ 25 ¾ÄÉÀÒ¤Á 2009 21:38 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 3 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ -InnoXenT- à˵ؼÅ: ¾ÔÁ¾ì¼Ô´ T T |
#2
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ÍéÒ§ÍÔ§:
áÅéÇ¡çâ´Â»¡µÔ ¤×Íã¹·Ò§¤³ÔµÈÒʵÃì àÃÒäÁè¶×ÍÇèÒ $\infty$ à»ç¹µÑÇàÅ¢ ´Ñ§¹Ñé¹á·¹·Õè¨Ðà¢Õ¹ $A_{\infty}$ à¢Õ¹ÇèÒ $\displaystyle\lim_{n\to\infty}A_n$ ¨Ð´Õ¡ÇèÒ¤ÃѺ
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#3
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¶éÒ¨ÐãËé¼Áá¡éä¢ ¼Á¡ç§§æµÑÇàͧÍÂÙèàËÁ×͹¡Ñ¹ÇèÒ ¾ÔÁ¾ìÍÍ¡ÁÒä´éä§
¶éÒÊÁÁµÔÇèÒ $A_n$ à»ç¹ÅӴѺ¢Í§Í¹Ø¡ÃÁ͹ѹµì â´Â·Õè $A_{n+1} = \frac{2}{3}A_n - \frac{1}{3^{2n-1}}$ àÁ×èÍ $n \geqslant 1$ ¨ÐàÇÔÃ줡ÇèÒÁÑê¤ÃѺ ÊèǹàÃ×èͧ lim ¼Áá¡éä¢áÅéÇ
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àÁ×èÍäÃàÃÒ¨Ðà¡è§àÅ¢¹éÒÒÒÒÒÒ ~~~~ T T äÁèà¡è§«Ñ¡·Õ ·Óä§´Õ 25 ¾ÄÉÀÒ¤Á 2009 21:38 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ -InnoXenT- |
#4
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ÍéÒ§ÍÔ§:
áÅéÇ¡ç¤èÒ·Õèá¹è¹Í¹¢Í§ $A_1$ ËÁÒ¤ÇÒÁÇèÒÂѧ䧤ÃѺ ¡ç... ÅӴѺàÃÒÁѹàÃÔèÁ´éÇÂÍÐäáçä´é $A_1$ ¡çà»ç¹ÍÐäáçä´éäÁèãªèàËÃͤÃѺ ËÃ×ÍÇèÒ¼Áà¢éÒ㨼Դàͧ
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#5
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¨ÐÇèÒ仵͹¤Ô´ ¡çÅ×Á¹Ö¡¶Ö§¤èÒ $A_1$ áÎÐ
¢Íâ·É·Õè·ÓãËéàÊÕÂàÇÅÒ¤ÃѺ
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àÁ×èÍäÃàÃÒ¨Ðà¡è§àÅ¢¹éÒÒÒÒÒÒ ~~~~ T T äÁèà¡è§«Ñ¡·Õ ·Óä§´Õ |
#6
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ÍéÒ§ÍÔ§:
¨ÃÔ§æáÅéÇÊÒÁÒöËÒÊٵ÷ÑèÇ仢ͧ $A_n$ ä´é´éÇ«éÓ áµè¶éÒÍÂÒ¡ÃÙéá¤èÅÔÁÔµ¨ÐÑä´éÇèÒ $\displaystyle{\lim_{n\to\infty}A_n}=0$
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site:mathcenter.net ¤Ó¤é¹ |
#7
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ÍéÒ§ÍÔ§:
¤×Í ¼ÁÁÕ˹ѧÊ×Í ·ÕèÁÕà¹×éÍËÒàÃ×èͧ¹ÕéàËÁ×͹¡Ñ¹ áµè·ÓäÁè¤èÍÂä´éàŤÃѺ
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àÁ×èÍäÃàÃÒ¨Ðà¡è§àÅ¢¹éÒÒÒÒÒÒ ~~~~ T T äÁèà¡è§«Ñ¡·Õ ·Óä§´Õ |
#8
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ãËé $r,s\in (-1,1)$ áÅÐ $c\in\mathbb{R}$
¹ÔÂÒÁ $A_{n+1}=rA_n+cs^n$ ¨Ðä´é $A_{n+1}=rA_n+cs^n$ $~~~~~~~=r^2A_{n-1}+c(s^n+rs^{n-1})$ $~~~~~~~=\cdots$ $~~~~~~~=r^nA_1+c(s^n+rs^{n-1}+\cdots + r^{n-1}s)$ $~~~~~~~=r^nA_1+cs(s^{n-1}+rs^{n-1}+\cdots +r^{n-2}s+r^{n-1})$ ¶éÒ $r=s$ ¨Ðä´é $~~~~~~~ A_{n+1}=r^nA_1+cnr^n$ $~~~~~~~~~~~~~~=r^n(A_1+cn)$ ¶éÒ $r\neq s$ ¨Ðä´é $~~~~~~~ A_{n+1}=r^nA_1+cs\Big(\dfrac{s^n-r^n}{s-r}\Big)$ ·Ñé§Êͧ¡Ã³Õ¨Ðä´é $\displaystyle{\lim_{n\to\infty}A_n}=0$ ÂÔ觡ÇèÒ¹Ñé¹ $\displaystyle{\sum_{n=1}^{\infty}A_n}$ ¡çËÒ¤èÒä´é
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site:mathcenter.net ¤Ó¤é¹ 26 ¾ÄÉÀÒ¤Á 2009 01:57 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nooonuii |
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