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·í§Ú°ª¤¤ªº®É­Ô¡A¥¿­È©Ò¿×¡u·s¼Æ¾Ç¡v¶}©l¤§»Ú¡C¡u·s¼Æ¾Ç¡v·½©ó¬ü°ê¡A·í®ÉĬ«X²Ä¤@Áû¤H³y½Ã¬P¤ÉªÅ¡A°ê»Ú¤W¬°¤§¾_Å夣¤w¡A¬ü°ê¤H¬£»º³X°Ý¹Î­u«X¡A¬ã¨s«X°ê¦ó¥H¯à¥H¤@ÅK¹õ°ê®a¦Ó¦b¤ÓªÅ¬ã¨s¤è­±»â¥ý¡C¥L­Ì±o¨ìªºµ²½×«Ü¦h¡A¨ä¤¤¦³¤@ÂI«ü¥X¡A¬ü°ê¦b¬ì¾Ç¤è­±ªº°ò¦±Ð¨|¸¨¥î¡A±Ð§÷³¯Â¤£¼Å¹ê»Ú¡A©ó¬O¦U³B®i¶}½Òµ{¸ÕÅç¡A¬ã¨s¨Ã¥Xª©·sªº±Ð§÷¡C§Ú°ê¨ü¦¹­·¤§¼vÅT¡A§ï­²¹E°_¡C§Ú¤W°ª¤¤ªº®É­Ô¡A¡u·s¼Æ¾Ç¡v¤~¶}©l¤£¤[¡A©Ò±Ä¨úªº±Ð§÷¤j­P¤W¨ú¦Û SMSG (School Mathematics Study Group)¡A¨ä¯S¦â¬O¤Þ¶i³o¥@¬ö©Î«e¥@¬ö¤~µo®i¥X¨Óªº·s²z½×¡A¦p¶°¦X¡BÅÞ¿è¡Bªñ¥@¥N¼Æµ¥¡F³Ì­«­nªº¤@ÂI¬O¡A¤£¦A¥ú­«µøÁc½Æªº­pºâ¥H¤Î¸ÑÃøÃDªº¦U§O§Þ¥©¡A¥N¤§¥H²z½×ªº¤@³e©Ê¤ÎÄY®æ©Ê¬°µÛ²´ÂI¡C

­n¨D¼Æ¾Ç¤WªºÄY±K¦Û¦³¨ä²`»·ªº¾ú¥v­I´º¡C¼Æ¾Ç¤W¨ì³B¥Rº¡ªº³´¤«¨Ï¤H­Ì¤£±o¤£±Ä¨úÄYÂÔªººA«×¡C¦ý¬O¡A¹LµS¤£¤Î¡A¹L¤À±j½ÕÄY±K¤@µü¡A©¹©¹¨Ïªì¾ÇªÌ¤£¯à»â·|¡A¬Æ¦Ü±æ¦Ó¥Í¬È¡C°£«D§Ú­Ì¹ï©ó©Ò­n°Q½×ªº¨Æª«¥»¨­¤w¸g¥R¥÷ÁA¸Ñ¡A§_«hÄY±K±À²z¨S¦³¥²­n¡A³o¼Ë°µ¤£¦ý¤£¯àÅý§Ú­Ìª¾¹D§ó¦h¡A¤Ï¦Ó¬O¤@¤jªýê¡C¨Æ¹ê¤W¡A¼Æ¾Ç¥»¨­¦³³\¦h¬O¹³ª«²z¤@¼Ë¡A»Ý­n¾a¸gÅç¡A³z¹L³\¦h¯S¨Ò¡A¤@¦A¤Ï¬M¥X¨ä¥»½è¥H«á¡A©â¶H¤~¬O¤@­Ó²¼äªº¤u¨ã¡A§_«h¥²®{µMµL¥Î¡C

¦³®É­Ô¡A§Ú­Ì·|µo²{¼Æ¾Ç¤Wªº¤@¨ÇÃÒ©ú¨Æ½è¤W¤Q¤ÀµL¥Î¡CÁ|¨Ò¨Ó»¡¡A½Ò¥»¤W¥s§Ú­ÌÃÒ©ú

\begin{displaymath}
1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}
\end{displaymath}

³o¼Ëªº¦¡¤l´X¥G¬O²³©Ò©Pª¾¡A¦Ó¥B¬O±`Ãѯ몺»Ý­n°O¦í¡C §ÚÁÙ°O±o¤Q¤À²M·¡¡A°ª¤¤®Éªº¦Ñ®vªá¤F¦nªø¤@¬q®É¶¡¡A«Ü½æ¤Oªº¸ÑÄÀ¼Æ¾ÇÂk¯Çªk¡A¦Ó¥B¤@¦A±j½Õ¡A°£¤F¥Î¼Æ¾ÇÂk¯Çªk¡Aµ´¹ï¤£¯à¡uÄY®æ¡vªºÃÒ©ú³o­Ó¦¡¤l¦¨¥ß¡Cª½¨ì«Ü¤[¥H«á¡A§Ú­Ì¤j¦h¼Æ¤HÁÙ¬O¤£¯à¤Q¤ÀÁA¸Ñ¼Æ¾ÇÂk¯Çªkªº¥»½è¡C§Ú­Ì¥u¬O²ßºDªº¥ý¥Î n=1 ¥N¤J¡AÅçÃÒ¨S¿ù¡A¦A±q n ¦¨¥ß¡AÃÒ©ú n+1 ¦¨¥ß¡A³oµL«D¬O¦b¦¡¤l¨âºÝ¦U¥[ (n+l)2¡A¦A§â¥kÃä´ê¦¨¬J©w®æ¦¡¡A©ó¬O®Ú¾Ú¼Æ¾ÇÂk¯Çªk¡A±oÃÒ¡C§Ú­Ó¤H´¿¸gªá¤F«Üªø¤@¬q®É¶¡¥h«ä¦Ò³o­Ó°ÝÃD¡A¦Û¤v°Ý¦Û¤w¡A¦pªGÃD¥Ø¬O $1^2+2^2+\cdots+n^2=?$ ¨º¸Ó«ç»ò¿ì¡H¬Æ¦Ü¦pªG±N¥­¤è§ï¦¨ m ¦¸¤è¡A°Ý $1^m+2^m+\cdots+n^m=?$ ¤S¸Ó¦p¦ó¡H¸g¹L«Üªøªº¤@¬q±Ã¤ã¡A§Ú¤£±o¤£©Ó»{¡uÃÒ©úµL¥Î½×¡v¡A§Ú©Ò¥H¦p¦¹»¡¡A¨Ã«D¤£¬Û«H¨º¨ÇÃÒ©ú¡A¥u¬O²`²`ªºÄ±±o¡A¦b§Ú­Ì¯u¥¿ÁA¸Ñ°ÝÃD¥»½è¡A¤]´N¬O¡A¦b§Ú­Ì¯àÄY±KÃÒ©ú¤§«e¡A§Ú­Ì¥²¤w¥¿¦¡©Î«D¥¿¦¡ªº«ä¯Á¹L¥¦¡A±o¨ì®t¤£¦h¨¬°÷ªºÃÒ¾Ú¡A¦Ü©óÃÒ©ú¥»¨­¡A¦³®É«J´N¹³µ²±BÃҮѤ@¼Ë¡A¥Î¨Ó¨Ï¤H§ó¬Û«H¨Æ±¡ªº¯u¹ê©Ê½}¤F¡C¼Æ¾Ç¥»¨­¡A»Ý­n®üÃö¤ÑªÅªº«ä·Q¡A²§·Q¤Ñ¶}ªº³Ð³y¡A¥u¦³·í§A¦³¤F¤@¨Ç²Ê²¤ªº­ì©lºc·Q¥H«á¡AÅÞ¿è©MÄY®æ¤~¥Î¨Ó¨Ï²z½×¥»¨­§ó©ö©óÁA¸Ñ¡A©ÎªÌ»¡¬O´«­Ó¤è¦¡¥[¥HÅçÃÒ½}¤F¡C

³oùØ­nÁ|ªº¤@­Ó¨Ò¤l¡A¬O¤×©Ô (Euler) ­pºâ $\frac{1}{1^2} + \frac{1}{2^2} + \cdots + \frac{1}{n^2} + \cdots$ ªº¤èªk¡C¨Ã¨S¦³¦h¤Ö¤H¯à¦³¥L³o¼Ë²§·Q¤Ñ¶}ªº«ä·Q¡A¨Æ¹ê¤W¡A¤×©Ô¦b¼Æ¾Çªº³\¦h¨¤¸¨³£¦³¯S®íªº°^Äm¡A¥Lªº³\¦h¯«¨Ó¤§µ§¡A¦b§Ú­Ì¬Ý¨Ó³£·¥¨ä¡u®Çªù¥ª¹D¡v¡C

¬f§V§Q (Bernoulli) ¬O¤×©Ôªº¥ý½ú¡A¥L¹ïµL½a¯Å¼Æ·¥¦³°^Äm¡A²×¨ä¤@¥Í¡A³\¦h¯Å¼Æ³£³Q¥L¬ã¨s¹L¡AÄ´¦p¡AµÛ¦Wªº¬f§V§Q¯Å¼Æ´N¬O¦]¥L¦Ó¦W¡C¹ï©ó $\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}+\cdots=?$ ¬f§V§Q©l²×¥¼¯à­pºâ¥X¨ä­È¡A¹ï¦¹¡A¥L¼g¹D¡G¡u¦pªG¦³¤H¯à§i¶D§Ú³o­Ó¯Å¼Æ¨D©Mªº¤èªk¡A§Ú±N¤Q¤À·P¿E¡C¡v¬f§V§QÁöµM¤£¯à¿Ë¦ÕÅ¥¨£¦³¤H§i¶D¥L³o­Ó¯Å¼Æªº¨D©M¤èªk¡A¦ý¤£¤[«á¡A¤×©Ô´Nµ¹¤F¤@­Ó·¥¨ä´I¦³¤Û·Qªº¸Ñµª¡C

­º¥ý¡A¤×©Ôª`·N¨ì¦h¶µ¦¡®Ú©M«Y¼ÆªºÃö«Y¡A¦pªG¦h¶µ¦¡

\begin{displaymath}
P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\quad (a_n\neq 0)
\end{displaymath}

ªº®Ú¬O r1, r2,¡K,rn¡A«h

\begin{displaymath}
\begin{eqalign}
P(x) &= a_n(x-r_1)(x-r_2)\cdots(x-r_n) \\
...
...ots \\
& \quad {} + (-1)^na_n(r_1r_2\cdots r_n)
\end{eqalign}\end{displaymath}

¤ñ¸û«Y¼Æ¥i¥H±o¨ì $r_1+r_2+\cdots+r_n=-\frac{a_{n-1}}{a_n}$¡A¦pªG¦h¶µ¦¡

\begin{displaymath}
q(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+a_{2n-2}x^{2n-2}+\cdots
a_0 \quad (a_{2n} \neq 0)
\end{displaymath}

ªº®Ú¦¨Âù¥H $\pm r_1,\pm r_2$,¡K,$\pm r_n$ ªº«¬¦¡¥X²{¡A«hÃþ¦ü¤Wªk

\begin{displaymath}
\begin{eqalign}
q(x) &= a_{2n}(x-r_1)(x+r_1)(x-r_2)(x+r_2)\c...
...ad {} + (-1)^n a_2n(r_1^2r_2^2r_3^2 \cdots r_n^2)
\end{eqalign}\end{displaymath}

¤ñ¸û«Y¼Æ¡A«h

\begin{displaymath}
a_{2n-1}=0 \quad , \quad r_1^2+r_2^2+\cdots+r_n^2=-\frac{a_{2n-2}}{a_{2n}}
\end{displaymath}

¦pªG q(x) ¨S¦³¹s®Ú¡A«h¦Ò¼{

\begin{eqnarray*}
q(\frac{1}{x})
&=& a_{2n}(\frac{1}{x})^{2n} + \cdots + a_2(\f...
...1}{x})^2
+ a_1(\frac{1}{x})+a_0 \\
&=& (\frac{1}{x})^{2n}Q(x)
\end{eqnarray*}


¨ä¤¤

\begin{displaymath}
Q(x) = a_0x^{2n}+a_1x^{2n-1}+a_2x^{2n-2} + \cdots
a_{2n-1}x+a_{2n} \quad (a_0\neq 0)
\end{displaymath}

¨ä®Ú«ê¬° q(x) ½Ñ®Úªº­Ë¼Æ¡A§Y $\pm \frac{1}{r_1}$, $\pm \frac{1}{r_2}$,¡K, $\pm \frac{1}{r_n}$¡A¥é·Ó«e½×¡A¥ç±o

\begin{displaymath}
a_1=0 \quad , \quad
\frac{1}{r_1^2}+\frac{1}{r_2^2}+\cdots+\frac{1}{r_n^2}=-\frac{a_2}{a_0}
\end{displaymath}

¤W­±¥kÃ䪺¦¡¤l©M©Ò­n¨Dªº $\frac{1}{1^2}+\frac{1}{2^2}+\cdots \frac{1}{n^2}+\cdots$ ¦³´X¤À¬Û¦ü¤§³B¡A¥u®t¤@­Ó¬O¦³­­¶µ¡A¤@­Ó¬OµL­­¶µ½}¤F¡C ¦Ò¼{ $\sin {x}$¡A§Q¥Î®õ°Ç®i¶}¦¡

\begin{displaymath}
\sin x = x -\frac{1}{3!}x^3+\frac{1}{5!}x^5 \cdots+(-1)^{2n+1}
\frac{1}{(2n+1)!}x^{2n+1}+\cdots
\end{displaymath}

¤×©Ô§â $\sin x$ µø¬° $n=+\infty$ ªº¦h¶µ¦¡¡A¨ä®Ú¬° $0,\pm\pi,\pm 2\pi$,¡K, $\pm n\pi,\cdots$¡C±N³o­Ó¡u¦h¶µ¦¡¡v°£¥H x¡A±o¨ì¡uµL½a¶µ¡v¦h¶µ¦¡

\begin{displaymath}
R(x)=1-\frac{1}{3!}x^2+\frac{1}{5!}x^4\cdots
\end{displaymath}

¨ä®Ú¬° $\pm\pi,\pm 2\pi$,¡K, $\pm n\pi,\cdots$ §¡¤£¬°¹s¡A©ó¬O§Q¥Î«e­zªº²z½×¡A±o¨ì

\begin{displaymath}
\frac{1}{(1\pi)^2}+\frac{1}{(2\pi)^2}+\cdots+\frac{1}{(n\pi)^2}+\cdots
= -\frac{a_2}{a_0}=\frac{1}{6}
\end{displaymath}

¥ç§Y

\begin{displaymath}
\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}+\cdots=\frac{{\pi}^2}{6}
\end{displaymath}

§Ú²q·Q¡A¦pªG¦b¦Ò¸Õ¨÷¤W³o¼Ë°µµªªº¸Ü¡A¤j¥bªº¥i¯à¬O±o¨ì¹s¤À¡C¤×©Ô¥»¤H¹ï³o¼Ëªº¸ÑªkÅãµM¤]¨Ã¤£º¡·N¡A¥L´¿¸Õ¹L¦UºØ¨ä¥L¥i¯à¡A·QÀò±o¤@­Ó¸û¥¿¦¡ªºÃÒ©ú¡FÁöµM¯u¥¿­pºâ¨ì¤p¼ÆÂI¥H«á´X¦ì¡AÅý¤H¦³´X¤À¬Û«H¨ä¥¿½T©Ê¡A¦ý¬O©Ò¦³¤H¡A¥]¬A¤×©Ô¦b¤º¡A³£ªáºë¯«¥h§ä§ó¦nªºÃÒ©ú¡C¤×©Ô¥»¤H§ó§Q¥ÎÃþ¦üªº¤èªk¡A±o¨ì³\¦h¨ä¥Lªº¦¡¤l¡C

§Ú·Q»¡ªº¤@ÂI¬O¡A¦³¤Fµª®×¡A¦A´êµÛ¥hÃÒ©ú¨ä¥¿½T©Ê¡A´NÅã±o¤ñ­ì¨Ó§ó®e©ö¡A§ó¦³ÀYºü¤F¡C¦b³oùاڷQ´£¨Ñ¤@­Ó°ò¥»ªº¤èªk¡A¶È»ÝÀ´±o¤@¨Ç¤T¨¤ªº­pºâ´N°÷¡A¦Ó¥B¾A·íªº­×¹¢¥H«á¡A§ó¥i¥H¨D±o¤@¯ë $P(2k)=\frac{1}{1^{2k}}+\frac{1}{2^{2k}}+\cdots
+\frac{1}{n^{2k}}+\cdots$ ¤§­È¡C ¦b³o½gµu¤åùؤ£´£¤Î¦¹Ác½Æªº­pºâ¡A¦³¤@ÂI«Ü¦³½ìªº¬O¡A³o¤@¯ë¦¡¤lªº©M»P¬f§V§Q ¯Å¼Æ·¥¦³Ãö«Y¡C

©M¤×©Ôªº¸Ñªk¬Û¤Ï¡A§Ú­Ì·Q¥Î $\pm \frac{1}{k\pi}$ ¥»¨­¨Óºc¦¨¦h¶µ¦¡ªº®Ú¡A¦Ó«D $\pm k\pi$¡C·íµM³o¤¤¶¡¦³§xÃø¦s¦b¡A§Q¥Î¤T¨¤¨ç¼Æ§Ú­Ì¥i¥H¶°éÁ×¹L³o­Ó§xÃø¡C

¥Ñ De Moivre ©w²z¥i±o

\begin{displaymath}
(\cos{\theta}+i \sin{\theta})^{2n+1}=\cos{(2n+1)\theta}+i \sin{(2n+1)\theta}
\end{displaymath}

¦pªG¿ï©w¯S§Oªº¨¤«× $\theta=\pm \frac{\pi}{2n+1}$, $\pm \frac{2\pi}{2n+1}$, ¡K, $\pm \frac{n\pi}{2n+1}$ ¡]§¡¬°¾U¨¤¡^¡A«h

\begin{displaymath}
(\cos{\theta}+i \sin{\theta})^{2n+1}=\cos{k\pi}+i \sin{k \pi}=\pm 1
\end{displaymath}

¥t¤@¤è­±§Q¥Î¤G¶µ¦¡©w²z®i¶}¥i±o

\begin{displaymath}
\begin{eqalign}
\lefteqn{ (\cos{\theta}+i \sin{\theta})^{2n+...
...1}_3 (\cos{\theta})^{2n-2}(\sin{\theta})^3+\cdots
\end{eqalign}\end{displaymath}

¤ñ¸û¨â¦¡ªºµê¼Æ³¡¥÷¡A¥i¥H±o¨ì

\begin{displaymath}
\begin{eqalign}
& C^{2n+1}_1(\cos{\theta})^{2n}\sin{\theta}-...
...1}_5 (\cos{\theta})^{2n-4}(\sin{\theta})^5-\cdots
\end{eqalign}\end{displaymath}

µ¥©ó 0¡C¦b¤W­±ªº¯S§O¿ï©w¨¤¡A $\sin{\theta}\neq 0$¡A¥i¥H±N¤W¦¡°£¥H $(\sin{\theta})^{2n+1}$¡A±o¨ì

\begin{displaymath}
C^{2n+1}_1(\cot{\theta})^{2n}-C^{2n+1}_3 (\cot{\theta})^{2n-2}
+C ^{2n+1}_5 (\cot{\theta})^{2n-4}-\cdots=0
\end{displaymath}

´«¥y¸Ü»¡¡A¦h¶µ¦¡

\begin{displaymath}
P(x)=C^{2n+1}_1 x^{2n}-C^{2n+1}_3 x^{2n+2}+C^{2n+1}_5 x^{2n+4}-\cdots
\end{displaymath}

ªº 2n ­Ó«D¹s®Ú¬° $\pm \cot\frac{\pi}{2n+1}$, $\pm \cot\frac{2\pi}{2n+1}$,¡K, $\pm \cot\frac{n\pi}{2n+1}$¡A¦Ó®Ú»P«Y¼ÆªºÃö«Y¦A«×«OÃÒ

\begin{displaymath}
\begin{eqalign}
& (\cot{\frac{\pi}{2n+1}})^2+(\cot{\frac{2\...
...+1}_3}{C^{2n+1}_1} = \frac{n(2n-1)}{3}
\end{eqalign}\eqno{(*)}
\end{displaymath}

±q¥t¤@¨¤«×¦ô­p¾U¨¤©M¨ä¤T¨¤¨ç¼Æ­È¶¡ªº»~®t¡A¥i¥H±o¨ì

\begin{displaymath}
\sin{\theta}<\theta<\tan\theta,\quad (0<\theta<\frac{\pi}{2})
\end{displaymath}

³o¤@ÂI¥i¥Ñ¤U¹Ï³æ¦ì¶ê¤º®°§Î­±¿n¬Ý¥X¨Ó¡G



$\triangle OAB$ ¤§­±¿n < ®°§Î OAB ¤§­±¿n < $\triangle OAC$ ¤§­±¿n

\begin{eqnarray*}
& \frac{1}{2}\sin\theta < \frac{1}{2}\theta < \frac{1}{2}\tan\theta \\
& \sin\theta < \theta < \tan\theta
\end{eqnarray*}


¥­¤è«á¡A¦A­Ë¼Æ¡A±o

\begin{displaymath}
{\csc}^2\theta > \frac{1}{{\theta}^2}>{\cot}^2\theta \quad i.e. \quad
1+{\cot}^2\theta>\frac{1}{{\theta}^2} > {\cot}^2\theta
\end{displaymath}

±N¤W¦¡¥Î $\theta=i \frac{\pi}{2n+1}$ $(i=1,2,\cdots,n)$ ¥N¤J¡A¨Ã±N n ­Ó¤£µ¥¦¡¬Û¥[¡A±o¨ì

\begin{displaymath}
\begin{eqalign}
& n+(\cot{\frac{\pi}{2n+1}})^2+\cdots+(\cot...
...\pi}{2n+1}})^2+\cdots+(\cot{\frac{n\pi}{2n+1}})^2
\end{eqalign}\end{displaymath}

¨Ã¥Ñ(*)±o

\begin{displaymath}
n+\frac{n(2n-1)}{3}>(\frac{2n+1}{\pi})^2(\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2})>(\frac{n(2n-1)}{3})
\end{displaymath}

¥ç§Y

\begin{displaymath}
\frac{{\pi}^2(2n^2+n)}{3(2n+1)^2}>\frac{1}{1^2}+\frac{1}{2^2}+\cdots\frac{1}{n^2}>\frac{{\pi}^2n(2n-1)}{3(2n+1)^2}
\end{displaymath}

·í $n\rightarrow\infty$ ®É

\begin{displaymath}
\frac{{\pi}^2(2n^2+n)}{3(2n+1)^2}\rightarrow\frac{{\pi}^2}{6...
...d \frac{{\pi}^2n(2n-1)}{3(2n+1)^2}\rightarrow\frac{{\pi}^2}{6}
\end{displaymath}

©Ò¥H

\begin{displaymath}
\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}\rightarrow\frac{{\pi}^2}{6}
\end{displaymath}

¦]¦¹

\begin{displaymath}
\frac{1}{1^2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}+\cdots=\frac{{\pi}^2}{6}
\end{displaymath}

[«áµù]¡m¼Æ¾Ç¶Ç¼½¡n©u¥Z²Ä¥|´Á¸­©Û©w¥ý¥Í¡q±Æ®e­ì²z¡r¤@¤å¡A²Ä86­¶¡A¦³¤@¦³½ìÃD¥Ø¡G ¡u³] n ¬O¦ÛµM¼Æ¡A¤G¾ã¼Æº¡¨¬ $1\leq a$,$b\leq n$¡Fsn ªí¥Ü a,b ¤¬½èªº¾÷²v¡A¸ÕÃÒ $\lim_{n\rightarrow\infty}s_n=s$ ¦s¦b¡C¡v¨Æ¹ê¤W $\lim_{n\rightarrow\infty}\,s_n$ ¤£¦ý¦s¦b¡A¦Ó¥B

\begin{eqnarray*}
\lim_{n\rightarrow\infty}\,s_n&=&\lim_{n\rightarrow\infty}(1-\...
...rac{1}{p_1^2})(1-\frac{1}{p_2^2})\cdots(1-\frac{1}{p_n^2})\cdots
\end{eqnarray*}


±q¥t¤@­Ó¨¤«×¬Ý

\begin{eqnarray*}
\frac{1}{(1-\frac{1}{p_1^2})}&=&1+\frac{1}{p_1^2}+\frac{1}{p_1...
...})}&=&1+\frac{1}{p_n^2}+\frac{1}{p_n^4}+\cdots\\
\vdots&&\vdots
\end{eqnarray*}


±N¦U¦¡¬Û­¼¡]¦A¥[¥H·¥­­ªº·§©À¡^±o¨ì³q¶µ¬° ${\displaystyle \frac{1}{p_{i_1}^{2n_1}p_{i_2}^{2n_2} \cdots p_{i_m}^{2n_m}} }$¡A ¨C¤@¶µ§¡¬O $\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots\frac{1}{n^2}$ ¤§¤@¶µ¡A¤Ï¤§«á¦¡ªº¥ô¤@¶µ¤]¥i¥Hªí¥Ü¦¨«eªÌªº³q¶µ§Î¦¡¡C¡]§Q¥Îºâ³N°ò¥»©w²z¡A¨C­Ó¦ÛµM¼Æ¥i¥H°ß¤@¤À¸Ñ¦¨½è¦]¼Æ­¼¿n¡C¡^©Ò¥H

\begin{displaymath}
\frac{1}{1-\frac{1}{p_1^2}}\cdot\frac{1}{1-\frac{1}{p_2^2}}\...
...2}+\frac{1}{2^2}+\cdots+\frac{1}{n^2}+\cdots=\frac{{\pi}^2}{6}
\end{displaymath}

­Ë¼Æ±o $\lim_{n\rightarrow\infty} \; s_n=\frac{6}{{\pi}^2}$¡A¬O¬°¥¿½T­È¡C

[²ßÃD] §Q¥Î¦UºØ¤èªk¡A¥¿¦¡©Î«D¥¿¦¡ªº¡A¨D $\frac{1}{1^4}+\frac{1}{2^4}+\cdots+\frac{1}{n^4}\cdots$ ¤§©M µù ¡C

[´£¥Ü¡G $x^4 - \frac{1}{a^4}=(x-\frac{1}{a})(x+\frac{1}{a})(x-\frac{i}{a})
(x+\frac{i}{a})$]

 
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