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#31
01 ÁԶعÒ¹ 2006, 23:40
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¨ÃÔ§æ¨Ò¡ ¡ÒáÃШÒ¢ͧ¹éͧ ¡ç·ÓµèÍä´éäÁèÂÒ¡´éÇÂÇÔ¸Õ Á.»ÅÒÂáÅéǤÃѺ
Êèǹ¶éҨзÓẺ recurrence ¡è͹¡çµéͧá¡éÊÁ¡ÒÃÍÍ¡ÁÒ¤ÃѺ ãËé $ a_n = \alpha^n $ ᷹ŧä»ã¹Êèǹ·Õè recurrence áÅéÇá¡éÊÁ¡Òà ËÒ $\alpha $ ¡ç¨Ðä´é 5 ¡Ñº -1 ´Ñ§¹Ñé¹ ¤ÓµÍº¡Ã³Õ·ÑèÇ仢ͧ $ a_n $ ¤×Í $ A(5)^n + B(-1)^n $ «Ö觨ҡ 2 à·ÍÁáá á·¹¤èÒÅ§ä» áÅéÇá¡éÊÁ¡ÒÃËÒ¤èÒ A,B ¡ç¨Ðä´é ÅӴѺ·ÕèÊÁºÙóì¤ÃѺ
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#32
09 ÁԶعÒ¹ 2006, 14:13
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¢Í§¤Ø³ passer-by ÁÕÇÔ¸ÕÍ×è¹ÍÕ¡äËÁ¤ÃѺ
next $$\sum_{k=1}^\infty (\ln x)^k = \frac{\pi}{6-\pi}$$
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒ㨠
22 µØÅÒ¤Á 2006 13:17 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 2 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ Mastermander
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#33
09 ÁԶعÒ¹ 2006, 14:53
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à·ÍÁ´éÒ¹º¹¤×Í $$\sum_{k=1}^{\infty}\ (\ln x)^k=\frac{\ln x}{1-\ln x}=\frac{\pi}{6-\pi}$$
¡ÅѺàÈÉà»ç¹ÊèǹáÅéǺǡ·Ñé§Êͧ¢éÒ§´éÇÂ˹Öè§ ¨Ò¡¹Ñ鹤ٳä¢Çé¨Ðä´é $\ln x=\pi/6$ ´Ñ§¹Ñé¹ $x=\exp{(\pi/6)}$
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish. 09 ÁԶعÒ¹ 2006 15:25 : ¢éͤÇÒÁ¹Õé¶Ù¡á¡éä¢áÅéÇ 1 ¤ÃÑé§, ¤ÃÑé§ÅèÒÊØ´â´Â¤Ø³ nongtum
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#34
09 ÁԶعÒ¹ 2006, 15:00
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ÍéÒ§ÍÔ§:
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#35
09 ÁԶعÒ¹ 2006, 15:20
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ãªè¤ÃѺ
 µÒÁä»á¡éáÅéǹФÃѺ àÍÒ⨷ÂìÁÒá»ÐÍÕ¡¢éÍ´éÇÂàŨ§áÊ´§ÇèÒ $$\sum_{k=2}^{\infty}\ \frac{\ln{(1+\frac1k)}}{\ln (k^{\ln (k+1)})}=\frac1{\ln2}$$
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¤¹ä·ÂÃèÇÁã¨ÍÂèÒãªéÀÒÉÒÇÔºÑµÔ ½Ö¡¾ÔÁ¾ìÊÑÅѡɳìÊÑ¡¹Ô´ ªÕÇÔµ(¤¹µÍºáÅФ¹¶ÒÁ)¨Ð§èÒ¢Öé¹àÂÍÐ (¨ÃÔ§æ¹Ð) Stay Hungry. Stay Foolish.
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#36
09 ÁԶعÒ¹ 2006, 15:35
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ÍéÒ§ÍÔ§:
$$ \begin{array}{lcr} \frac{a_1}{6}+\frac{a_2}{6^2}+\frac{a_3}{6^3}+\cdots= S\cdots(1)\\ \frac{4a_1}{6}+\frac{4a_2}{6^2}+\frac{4a_3}{6^3}+\cdots= 4S \cdots(2)\\ \frac{5a_1}{6^2}+\frac{5a_2}{6^3}+\frac{5a_3}{6^4}+\cdots= \frac{5}{6}S \cdots(3) \\ \frac{2}{3}+\frac{a_3}{6^2}+\frac{a_4}{6^3}+\cdots= \frac{29}{6}S \cdots(2)+(3) \\ \frac{2}{3}+6(\frac{a_3}{6^3}+\frac{a_4}{6^4}+\cdots)= \frac{29}{6}S \\ \frac{2}{3}+6(S-\frac{1}{4}) =\frac{29}{6}S \end{array} $$ ºÃ÷Ѵ·Õè 2,3 à¡Ô´¨Ò¡àÍÒ 4 áÅÐ 5/6 ¤Ù³µÅÍ´ÊÁ¡Ò÷Õè 1 ¤ÃѺ Êèǹ¡ÒÃ¹Ó ÊÁ¡Òà (2) +(3) àÃҨкǡ¡Ñ¹ã¹á¹ÇàÂ×éͧæ àËÁ×͹·Õè¹éͧºÇ¡ã¹Í¹Ø¡ÃÁ Á.»ÅÒ¹Ñè¹áËÅФÃѺ áÅéÇ¡çãªé recurrence relation à¢éÒ仪èÇ ÊèǹºÃ÷Ѵ·ÕèÁÕ $ S-\frac{1}{4}$ ¡çÁÒ¨Ò¡ÊÁ¡Ò÷Õè 1 áÅéǨѺÂéÒ¢éÒ§¹Ñè¹àͧ¤ÃѺ ¨Ò¡¹Ñ鹡çá¡éÊÁ¡ÒÃËÒ¤èÒ S ¢éʹբͧÇÔ¸Õ¹Õé¤×Í äÁèãªé¤ÇÒÁÃÙéàÃ×èͧ¡ÒÃá¡éÊÁ¡Òà difference equation à¾×èÍãËéä´éÊٵ÷ÑèÇ仢ͧÅӴѺÍÍ¡ÁÒ
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#37
09 ÁԶعÒ¹ 2006, 16:08
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Hall algebra Problem
$$\frac{1^2}{3!}+\frac{2^2}{4!}+\frac{3^2}{5!}+...$$ 
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#38
09 ÁԶعÒ¹ 2006, 16:34
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ÊÓËÃѺ¤Ó¶ÒÁ¢Í§¤Ø³ Mastermander
$$ \sum_{n=1}^{\infty} \frac{n^2}{(n+2)!}= \sum_{n=1}^{\infty}\frac{1}{n!} -\sum_{n=1}^{\infty}\frac{3}{(n+1)!}+\sum_{n=1}^{\infty}\frac{4}{(n+2)!}=(e-1)-3(e-2)+4(e-\frac{5}{2})=2e-5 $$
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ
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#39
16 ÁԶعÒ¹ 2006, 23:10
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ÍéÒ§ÍÔ§:
¢é͵èÍä» 
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âÅ¡¹ÕéÁÕ¤¹ÍÂÙè 10 »ÃÐàÀ· ¤×Í ¤¹·Õèà¢éÒã¨àÅ¢°Ò¹Êͧ áÅФ¹·ÕèäÁèà¢éÒã¨
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#40
18 ÁԶعÒ¹ 2006, 16:19
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ÍéÒ§ÍÔ§:
à·èÒ·Õè¼Áà¤ÂàËç¹ Í¹Ø¡ÃÁẺ·ÕèµÔ´ difference equation ã¹ÃٻẺ¤ÅéÒÂæ⨷Âì·Õè¶ÒÁä» ¡çÊÒÁÒöãªéÇÔ¸Õ Á.»ÅÒÂÁÒ·´á·¹ä´é Êèǹ¤Ó¶ÒÁÅèÒÊØ´ ¨Ò¡ $ \frac{\arctan x}{x} = 1-\frac{x^2}{3}+\frac{x^4}{5}-\cdots $ áÅÐá·¹¤èÒ $ x= \frac{1}{\sqrt{3}}$ ¨Ðä´é¤ÓµÍºÍÍ¡ÁÒà»ç¹ $ \frac{\sqrt{3}\pi}{6} $ ·èÒ·Ò§¹éͧ Mastermander ¨Ð in love ¡ÑºÍ¹Ø¡ÃÁ͹ѹµìÁÒ¡¹Ð¤ÃѺ §Ñé¹Åͧ·Ó¢é͹Õé´ÙÁÑé Evaluate $$ \sum_{k=1}^{\infty}(-1)^{k+1}\frac{3k+1}{2k^3+k^2} $$ (Source: HMMT 2006)
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#41
19 ÁԶعÒ¹ 2006, 19:49
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¹Ò¹æ¨Ð·Óä´é·Õ¤ÃѺËØæ¢ÍµÑ´Ë¹éÒâ¾Ê·ì¡è͹ÅФÃѺ
 
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#42
20 ÁԶعÒ¹ 2006, 06:32
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¤ÓµÍº¢Í§¤Ø³ Timestopper_STG ¶Ù¡áÅéÇÅèФÃѺ
á¶ÁãËéÍÕ¡¢éÍ ¨Ò¡ hmmt 2006 àªè¹¡Ñ¹ Evaluate $$ \sum_{n=1}^{\infty} \frac{2n+5}{2^n(n^3+7n^2+14n+8)} $$
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à¡ÉÕ³µÑÇàͧ »ÅÒÂÁԶعÒ¹ 2557 áµè¨Ð¡ÅѺÁÒà»ç¹¤ÃÑ駤ÃÒÇ
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#43
21 ÁԶعÒ¹ 2006, 22:01
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µÍ¹ááàËÁ×͹¨Ðä´é§èÒÂæ¹Ð¤ÃѺä»æÁÒæàÅ¢äÁèŧàŤÃѺËÃ×ͼÁ·Ó¼Ô´???
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#44
21 ÁԶعÒ¹ 2006, 22:14
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àËÍæźàÈÉÊèǹ¼Ô´¨ÃÔ§æ´éǤÃѺáµè¼Áá¡é¡è͹â¾Ê·ìàÃÕºÃéÍÂÅФÃѺ*_*
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$$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x-b\sin x}{a\sin x+b\cos x}dx=\ln\left(\frac{a}{b}\right)$$ BUT $$\int_{0}^{\frac{\pi}{2}}\frac{a\cos x+b\sin x}{a\sin x+b\cos x}dx=\frac{\pi ab}{a^{2}+b^{2}}+\frac{a^{2}-b^{2}}{a^{2}+b^{2}}\ln\left(\frac{a}{b}\right)$$
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#45
22 ÁԶعÒ¹ 2006, 19:10
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¤ÓµÍº ¶Ù¡áÅéǤÃѺ
 ¶éÒã¤Ãä´é´Ùµé¹©ºÑº¢Í§¢éÍÊͺ hmmt ¢Í§ USA ·ÕèÊͺä»àÁ×èÍÃÒÇ ¡.¾.·Õè¼èÒ¹ÁÒ ¨Ð¾ºÇèÒ ¢éÍÅèÒÊØ´·Õè¼Á¶ÒÁä» ÁÕ¤Ðá¹¹àµçÁ¶Ö§ 15 ¤Ðá¹¹ Êèǹ¢éÍ¡è͹˹éÒ ÁÕ¤Ðá¹¹àµçÁ 18 ¤Ðá¹¹
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