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ÊÁѤÃÊÁÒªÔ¡ ¤ÙèÁ×Í¡ÒÃãªé ÃÒª×èÍÊÁÒªÔ¡ »¯Ô·Ô¹ ¤é¹ËÒ ¢éͤÇÒÁÇѹ¹Õé ·Óà¤Ã×èͧËÁÒÂÍèÒ¹·Ø¡ËéͧáÅéÇ

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Old 03 µØÅÒ¤Á 2016, 04:51
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ÊÁÁµÔãËé $(a_n)_n$ à»ç¹ÅӴѺ¢Í§¨Ó¹Ç¹¨ÃÔ§ºÇ¡ «Öè§ $$\sum_{n=1}^\infty a_n = \infty, \lim_{n \rightarrow \infty} a_n = 0$$ áÅéÇÊÓËÃѺ·Ø¡¨Ó¹Ç¹¨ÃÔ§ºÇ¡ $a$ ã´æ ¨ÐÁÕÅӴѺÂèÍ $(a_{n_k})$ «Öè§ $$\sum_{k=1}^\infty a_{n_k} = a.$$

¼ÁÃÙéÊÖ¡à¢éÒ㨵ÑÇ·Äɮպ·¹Ð¤ÃѺ àËÁ×͹ÇèÒµÑÇ $a_n$ ÁѹàÅç¡ÁÒ¡ áµè¶éҨѺÃÇÁ¡Ñ¹Áѹ¡çãË­èÁÒ¡æ´éÇ á¹Ç¤Ô´¡ÒþÔÊÙ¨¹ì¹èÒ¨ÐẺÇèÒ ¾ÂÒÂÒÁ sum $a_n$ ãËéã¡Åé $a$ ÁÒ¡·ÕèÊØ´ áÅéÇãªéà·ÍÁàÅ硢ͧ $a_n$ µÍ¹»ÅÒÂæ à¢éÒ仺ǡà¾ÔèÁ¨¹¼ÅÃÇÁÅÙèà¢éÒËÒ a

·Õè¤Ô´äÇé¤×Í ¨Ð¾ÂÒÂÒÁËÒ finite sum ¢Í§ $a_n$ ãËéã¡Åéæ $a$ áÅéǨÐËÒà·ÍÁ»ÅÒ¢ͧ $a_n$ «Ö觹éÍ¡ÇèҾǡ $\frac{1}{2^n}$ à¾×èÍãËé¼ÅÃÇÁàÅç¡ áÅкǡ¡Ñ¹ã¡Åé $a$ ¾Í´Õ

áµèà¢Õ¹¾ÔÊÙ¨¹ìäÁèä´é¤ÃѺ ´ÙáÅéÇà´ÒÇèÒ¤ÅéÒ Riemann rearrangement for condotional serie áµè¡çäÁèàËÁ×͹ Åͧ¾ÂÒÂÒÁ¤Ô´áÅéÇáµèÁѹäÁèÍÍ¡¤ÃѺ

ú¡Ç¹ªèÇ´éǤÃѺ

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03 µØÅÒ¤Á 2016 04:54 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 6 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ B º ....
µÍº¾ÃéÍÁÍéÒ§ÍÔ§¢éͤÇÒÁ¹Õé
  #2  
Old 04 µØÅÒ¤Á 2016, 21:34
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àª×èÍÇèÒãªé¤ÇÒÁÃÙéá¤è¹ÔÂÒÁ¢Í§ÅÔÁÔµ·Óä´é¹Ð
ÊÔ觷Õè¤Ø³µéͧ·Ó¤×Íá¤è·Ó¤ÇÒÁ¤Ô´¢Í§¤Ø³ãËéà»ç¹ÃÙ»¸ÃÃÁ

Åͧ¤Ô´´Ù´ÕæÇèÒÊÔ觷Õè¤Ø³µéͧ¾ÔÊÙ¨¹ìÁÕÍÐäúéÒ§

(1) ¹ÔÂÒÁ algorithm ¢Í§¡ÒÃÊÃéÒ§ÅӴѺ¢Í§ $n_k$ ·Õè¤Ø³¤Ô´ÇèÒãªéä´é¢Öé¹ÁÒ

(2) ¾ÔÊÙ¨¹ìÇèÒÅӴѺ¹Ñé¹ÊÍ´¤Åéͧ¡Ñº¢éͤÇÒÁ
for large enough $M$ and for any $e>0$, $\displaystyle a-\sum_{k=1}^M a_{n_k}<e$

ÊÔ觷ÕèÍÒ¨¨ÐªèǤس㹡ÒþÔÊÙ¨¹ìÁÕ

(1) $\displaystyle \sum_{k=m}^\infty a_k=\infty$

(2) for large enough $n$ and for any $e>0$, $a_n<e$
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µÍº¾ÃéÍÁÍéÒ§ÍÔ§¢éͤÇÒÁ¹Õé
  #3  
Old 05 µØÅÒ¤Á 2016, 09:15
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ãºé¡ÒÃÊÃéÒ§ÅӴѺÂèÍÂãËéÍÕ¡«Ñ¡¹Ô´ä´éÁÑé¤ÃѺ ãªé infremum ÁÑé¤ÃѺ Ẻ

Since $a_n \rightarrow 0$, there exists $i$ such that $a_i < a$. Choose $a_{n_1}$ such that $a_{n_1} < a , a - a_{n_1} < a - a_j$ for any $j$ with $a_j < a$. Since $a_{n_1} < a$, choose $n_2 > n_1$ such that $a_{n_2} < a - a_{n_1}$ and $ (a - a_{n_1}) - a_{n_2} < (a - a_{n_1}) - a_j$ for any $j > n_1$ and $a_j < (a - a_{n_1})$. Repeat the process.

Get $(a_{n_k})$ that each $a_k$ is the best closed to $a - \sum_{i=1}^{k-1} a_{n_i}$.

áµèÁѹ¹èҨеéͧàÅ×Í¡ãËé¢Ö鹡Ѻ $\epsilon$ à¾ÃÒÐ äÁè§Ñé¹äÁèÃÙé¨ÐáÊ´§Âѧä§ÇèÒÅӴѺÂèÍÂÅÙèà¢éÒÊÙè $a$ ¨ÃÔ§

á¹Ð¹Ó¡ÒÃÊÃéÒ§ÅӴѺ˹èͤÃѺ
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05 µØÅÒ¤Á 2016 09:18 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 5 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ B º ....
µÍº¾ÃéÍÁÍéÒ§ÍÔ§¢éͤÇÒÁ¹Õé
  #4  
Old 05 µØÅÒ¤Á 2016, 15:59
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à¢Õ¹ÍÍ¡ÁÒà»ç¹ algorithm ´ÕæÊÔ¤ÃѺ ÍÂèÒ§ÇԸբͧ¤Ø³ÊÒÁÒöà¢Õ¹ä´é´Ñ§¹Õé

1. àÅ×Í¡ $n_1$ à»ç¹¨Ó¹Ç¹·Õè¹éÍ·ÕèÊØ´ «Öè§ $a_{n_1} < a$ (àÅ×Í¡ä´éà¾ÃÒÐ $\displaystyle \lim_{n \rightarrow \infty} a_n = 0$)
2. ¶éÒÁÕ $n_1,n_2,...,n_k$ áÅéÇ«Öè§ $\displaystyle \sum_{i=1}^k a_{n_i} < a$ ¨ÐàÅ×Í¡ $n_{k+1}$ à»ç¹¨Ó¹Ç¹·Õè¹éÍ·ÕèÊØ´ÁÒ¡¡ÇèÒ $n_k$ áÅÐ $\displaystyle \sum_{i=1}^k a_{n_i}+a_{n_{k+1}}<a$ (àÅ×Í¡ä´éÍÕ¡à¾ÃÒÐ $\displaystyle \lim_{n \rightarrow \infty} a_n = 0$ àªè¹¡Ñ¹)

algorithm ¹Õéà»ç¹ algorithm ·Õè¶Ù¡µéͧáÅéǤÃѺ

áµèÊѧࡵÇèÒÂѧäÁèä´éãªéÍÕ¡¢éÍÁÙÅ˹Öè§àÅ ($\displaystyle \sum_{i=m}^\infty a_i=\infty$)
¤ÃÒǹÕé¢Ñ鹵͹µèÍ仨ÐàÍÒÁÒàª×èÍÁ¡Ñº $\epsilon$ Âѧ䧤ÃѺ
Åͧ¤Ô´àͧ¡è͹áÅéǤèÍÂà»Ô´ hint ¡çä´é¤ÃѺ
Ps. àÇÅҤԴ⨷ÂìẺ¹Õéá¹Ð¹ÓãËé¤Ô´¡è͹ÁÒµéͧÇҧἹ proof Âѧ䧺éÒ§æ ¡ÒäԴ algorithm àÃÔèÁ¨Ò¡¤Ô´¡è͹ÇèÒÍÂÒ¡·ÓÍÐäÃæ áÅéǶéÒÁѹµéͧãªéà¤Ã×èͧÁ×ÍÍÂèÒ§ infremum Áѹ¨ÐµÒÁÁÒàͧ¤ÃѺ
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05 µØÅÒ¤Á 2016 16:02 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Thgx0312555
µÍº¾ÃéÍÁÍéÒ§ÍÔ§¢éͤÇÒÁ¹Õé
  #5  
Old 08 ¸Ñ¹ÇÒ¤Á 2016, 18:23
kongp kongp äÁèÍÂÙèã¹Ãкº
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àÍÒẺãªé ·º.à«çµ ÃÖà»ÅèÒ ! ¡ÅØèÁ¢Í§ÊÁÒªÔ¡·ÕèÁÕà§×è͹䢴ѧ¹Õé ! »ÃСͺ´éÇ áÅÐ ææææ

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