·s¼W¦³Ãö¹ê¼Æ,µL¯Å¼Æªº§ó§¹¾ã¸ÑÄÀ. ³o¬O¥Ø«eª©¥», Åwªï§¹¾ã½Æ»sÂà¸ü.
---------------------------------------------------------------------------------------------------------
¦¹¤å¥Øªº¦b©ó«Ø¥ß¹q¸£ºâªk°ò¦
¦¹Àɸm©ó https://sourceforge.net/projects/cscall/files/MisFiles/RealNumber-zh.txt/download
¨Ã·|¤£®É§ó·s.
+------+
| ¹ê¼Æ |
+------+
n-¶i¨î©w㸃¼Æ::= ¥Ñn-¶i¨î¼Æ¦r¦ê©Òªí¹Fªº¼Æ. ¦¹¦r¦ê¥i§t¤@Ót¸¹©Î¤p¼ÆÂI:
<fixed_point_number>::= [-] <wnum> [ . <frac> ]
<wnum>::= 0 | <nzd> { 0, <nzd> }
<frac>::= { 0, <nzd> } <nzd>
<nzd> ::= (1, 2, 3, 4, 5, 6, 7, 8, 9) // '¼Æ¦r'·|ÀH¶i¨î§ïÅÜ
Ä´¦p: 78, -12.345, 3.1414159...(£k)
n-¶i¨î©w㸃¼Æªº¥[´îºâªk¦P¤p¾Ç±Ðªº(©Îºâ½L¤Wªº)ºâªk. ¥ô¨â¬Û¦Pn-¶i¨îªº¶i¨î
¼Æa,b ©Òªíªº¼Æ¬Û¦P iff a,bªº <fixed_point_number>ªí¹F§¹¥þ¬Û¦P(©Îa-b=0).
Ya¡Úb, «ha>b ©Î a<b, ¾Ü¤@¦¨¥ß (¤T¤@«ß).
A=B ::= 1. A¡ÝA // A¡ÝB «ü"ÄY®æ¦Pºc"
2. A¡ÝB ¡Û) 1-3/10^n = lim 0.999... =1
B= lim(n->¡Û) 1-2/2^n = lim 0.999... =1
C= lim(n->¡Û) 1-1/n = lim 0.999... =1
...
¥tºØ»¡ªk¬O 10*0.999... ³sÄò¼10¥u¯à¬Ý¨ì«e±ªº9,¬Ý¤£¨ì§À¬q¯u¥¿ªº¼Æ
µ²ºc. (10*0.999... ªº³B²z¤è¦¡§ïÅܤF¼Æµ²ºc. ¹ï©óµL¯Å¼Æ,³oÂI«Ü«n,
¦]µL¯Å¼Æ©Ò©w¸qªº¼Æ¥i¯à¦]¦¹¤v§ïÅÜ)
¥Ñ©ó<fixed_point_number> «D±`©ú½Tªº¯u¹ê¥B²o§è¨ìµL,¥ô¦³¦r²Õ¦¨ªº
²z½×µLªk§¹¥þ´yzℝ,'§¹³Æ'¬O¤£¥i¯àªº.
µù: ¦¹©w¸q隠§tªí©ú´`Àô¤p¼Æ爲µL²z¼Æ. ¶¶«K²µu»¡©ú±`¨£ªº¥N¼ÆÅ]³Nµý©ú:
(1) x= 0.999...
(2) 10x= 9+x // 10x= 9.999...
(3) 9x=9
(4) x=1
¸Ñµª: ¨S¦³¤½²z©Î©w²z¥iµý(1) => (2).
(2)¬O(1)ªºµL¦hºØ¸ÑÄÀ¤¤ªº¤@ºØ¸ÑÄÀ.
µù: §PÂ_´`Àô¤p¼Æx¬O§_爲¦³²z¼Æ¥i¥H³sÄòªº´î¥h´`Àô³¡¥÷p(i). ¦pªG¦b
¦³¨BÆJ¤º x-p(1)-p(2)-...=0, «hx爲¦³²z¼Æ,§_«h爲µL²z¼Æ. ¦]爲Yx爲
¦³²z¼Æ,³Ñ¾l³¡¥÷ r(i)= x-p(1)-p(2)-... ¥²¶·爲¤@´`Àô³¡¥÷p(i). ¦ý®Ú¾Ú
¡¥´`Àô¡¦©w¸q, r(i)¤£爲p(i). ¦]¦¹,´`Àô¤p¼Æ¤£爲¦³²z¼Æ(§YµL²z¼Æ).
©w²z: x∈ℚ,x>0 iff ¦s¦b¦³Óq,q∈ℚ, 0<q>0 ¥B,¦³Óq,...,¨Ï±ox=q1+q2+...
T F | F // x∈ℚ,x>0 ¥B,«D¦³Óq,...,¨Ï±ox=q1+q2+...
F T | F // x∉ℚ,x>0 ¥B,¦³Óq,...,¨Ï±ox=q1+q2+...
F F | T // x∉ℚ,x>0 ¥B,«D¦³Óq,...,¨Ï±ox=q1+q2+...
©w²z: ¦³ªøn-¶i¨î¼Æ©Òªí¼Æ爲¦³²z¼Æ. «D¦³ªøn-¶i¨î¼Æ©Òªí¼Æ爲µL²z¼Æ.
µý: ¥Dnª¬ªp¬O·í¼Æ爲µLªø¤p¼Æ. ¹ï¦¹,À³¥Î¥H¤W©w²z¥iµý.
¹ê¼Æ°ò¥»¤W²³æ´N¬O¼Æ¦r¥iµLªøªº¼Æ. ·¥¬O¥Î¨Ó©w¸q·L¤À,¤Î´£¨Ñ´M§ä¾É¼Æªº¤èªk,»P
¹ê¼Æ©w¸qµLÃö (Y¦³Ãö«Y,Ãþ¦ü¥H¤Wªº'ℝ'¤]n¥ý©w¸q,§_«h«ÜÃøÁ׶}¹PÀô½×µý°ÝÃD).
+------+
| ·¥ |
+------+
·¥::= lim(x->a) f(x)=L
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_04.pdf
http://www.math.ncu.edu.tw/~yu/ecocal98/boards/lec6_ec_98.pdf
https://en.wikipedia.org/wiki/Limit_(mathematics)
https://en.wikipedia.org/wiki/Limit_of_a_function
https://www.geneseo.edu/~aguilar/public/notes/Real-Analysis-HTML/ch4-limits.html
·¥ªººëÅè¦b©ó´£¨Ñ¤@ºØ¤£¸g¥Ñµ¥¦¡¨Ó´yz¼Æ(§Y,L)ªº¤èªk. ·¥·N«ä¬O»¡: xÁͪña
(x¡Úa)®É,f(x)ªº·¥¬OL (º¡¨¬£`-£_±Ôz), ¤£¬O"·íxÁͪñ©óa, ³Ì«áf(a)µ¥©óL".
Ä´¦p1: A= lim(n->¡Û) 1-1/n= lim(n->0⁺) 1-n= lim 0.999...=1
B= lim(n->¡Û) 1+1/n= lim(n->0⁺) 1+n= lim 1.000..?=1
Ä´¦p2: A=lim(x->ℵ₀) f(x), B=lim(x->ℵ₁) f(x) // ℵ₀,ℵ₁¬O§_«ê·í¬O¥tÓ°ÝÃD,¦ýY
¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@¡@// ±Ä¡§³Ì²×±N¬Û¦P¡¨¸ÑÄÀ,·|¦³°ÝÃD:
// f(ℵ₀)»Pf(ℵ₁)¬O§_¬Ûµ¥?
¡§·¥¡¨©w¸qA=B, ¦ý¤£¬O»¡·¥¤º®e¬Ûµ¥. Y±Ä"xÁͪñ..«hµ¥©ó.."»¡ªk, «h¦³«Ü¦h
ÅÞ¿è¤Wªº°ÝÃD.
µù:·¥¬O©w¸q¦b¤v¦sªº¼Æ¨t¤W,·¥µLªk¥Î¨Ó©w¸q¨ä©Ò¨Ï¥Îªº(Áͪñ§Ç¦C)¼Æ.
µù:·¥ªº¼ªk¤½¦¡(lim(x->c) (f(x)*g(x))= (lim(x->c) f(x))*(lim(x->c) g(x)) )
¥i¯à¦³ÂI°ÝÃD:
³]A=lim(n->¡Û) (1-1/n)= 1
A*A*..*A= ... = lim(n->¡Û) (1-1/n)^n // 1=1/e ?
µù:'µL¤p'¨Æ¹ê¤W¥i¯à¨Ã¤£¤p,¦]¨CÓ'¥i¦C¥X'ªºÁͪñ§Ç¦C©Òªí°Ï¶¡[x,c),¤jP¤W¨Ó»¡
,¤´»P¹ê¼Æℝ 1-1¹ïÀ³.
+------------------+
| (ÁÙì)·L¿n¤À¸àÄÀ |
+------------------+
http://www.math.ntu.edu.tw/~mathcal/download/precal/PPT/Chapter%2002_08.pdf
°²³]·L¿n¤À¬O¥Ñ¨Dºâ¨ç¼Æ±¿n°ÝÃD¶}©l: ³]F爲fªº±¿n¨ç¼Æ. ¥Ñ¨ç¼Æ±¿n¸q·N¥i±o:
(F(x+h)-F(x)) ¡Ü (f(x+h)+f(x))*(h/2) // h¬OÓ¨¬°÷¤pªº°¾²¾(´ú¸Õ)¶q
<F>0)¬Of(x)
F´Á±æ©Ê½è: (1)»~®t|lhs-rhs|ÀH·L¤p°¾²¾¶qhÄY®æ»¼´î(°Ñµù1) (2)h=0®É,lhs=rhs
¦]lhsªºh¤£¯à¬O0,©Ò¥H·L¿n¤Àªº°ò¥»°ÝÃD´N¬O´M¨D¯à¨Ï¥H¤W±Ôz¦¨¥ßªºF(©Îf)...
¬G,¤W±Ôz¥iªí¹F¦¨:
D(f(x))= lim(h->0) (F(x+h)-F(x))/h = f(x)
µù1: ½Ò¥»ªº»¡ªk¦³ÂI¤£¦P,¦ýF´Á±æ©Ê½èn¨D©Ò©w¸qªº¼Æ(§Y·¥ÈL)¶·°ß¤@,¨ä¥¦¤£ª¾¹D
µù2: §Æ±æ³oºØ¸ÑÄÀ¥i¼È®ÉÁ׶}¤£¥²nªºµL¤j(©Î¤p)ªº¸ÑÄÀ°ÝÃD,¶i¦Ó´£¨Ñ¤@¨Ç®¯½×
©Î²z½×¸û¥¿½Tªº¸ÑÄÀ°ò¦.
+----------+
| µL¯Å¼Æ |
+----------+
¯Å¼Æ::= S= £U(n=0,k) a(n)= a(0)+ a(1)+ a(2) +... +a(k)
a(n)ºÙ爲³q¶µ, a(0),a(1),...ºÙ爲¥[¶µ, nºÙ爲¯Á¤Þ, ¯Å¼ÆS爲¦U¥[¶µ¥Ñº¶µa(0)¥[¦Ü§À
¶µa(k)ªºÁ`©M. ¯Å¼ÆS«e(n<k)³¡¥÷¶µªºÁ`©MºÙ爲³¡¥÷©M. "a(0)+...+a(k)" ºÙ爲®i¶}¦¡.
µL¯Å¼Æ::= ·í¯Å¼Æ¦³µL¦hªº¥[¶µ®É(¯Á¤Þ§t¡Û),SºÙ爲µL¯Å¼Æ. ¤@¯ë¦Ó¨¥,µL¯Å¼Æªº
¥[Á`¹BºâµLªk¦b¦³¨BÆJ¤º§¹¦¨(³o¬O¡Ûªº¸q·N).
¦¬㪘::= µL¯Å¼Æªº³¡¥÷©M§Ç¦C¦³·¥.
µL¯Å¼Æ¹Bºâì«h: ®i¶}¦¡ªº³Ì«á¤@¶µ(¯Á¤Þ爲¡Ûªº¥[¶µ)¥²¶·¦C¥X,¥Hªí¥Ü¥[¶µµ²ºc.
®i¶}¦¡ªººâ³N»P¤@¯ë«DµL¯Å¼Æºâªk¬Û¦P:
¨Ò1: ³]S= £U(n=0,¡Û) a^n = 1+a+a^2+...+a^¡Û
S= 1+a*(1+a+a^2+...+a^¡Û)- a*a^¡Û
<=> S= 1+a*S-a^(¡Û+1)
<=> S(1-a)=1-a^(¡Û+1)
<=> S= (1-a^(¡Û+1))/(1-a)
¨Ò2: ³]S= £U(n=1,¡Û) n = 1+2+3+...+n
S= 1+2+3+...+n // (1)
S= n+...+3+2+1 // (2)
2S= n*(n+1) // (1)+(2)
<=> S= n*(n+1)/2
¤£¦C§À¶µ®É,®i¶}¦¡·|¦³«±Æ®Éªº'Å]³Nºâªk'°ÝÃD, ¦]®i¶}¦¡«±Æµ²ªG¥i¯à·|§ïÅÜì¯Å¼Æ
ªº©w¸q:
ÅP¦p1: S¥i¸g¥Ñ«±Æ¦Ó¦¨¥ô·N¼Æ:
S= £U(n=1,¡Û) n= 1+2+3+... =1+1+1+1+...= (1+1)+(1+1+1)+...
= £U(n=1,¡Û) n+1 // S©w¸q³Q§ïÅÜ
(©ÎªÌ S=(1+2)+(3+4)+... = £U(n=1,¡Û) 4*n-1)
ÅP¦p2:
S=1+2+4+8+... // ©¿²¤§À¶µ(¯f¦¡)
<=> S=1+2*(1+2+4+8+...) // ¦hºØ¥i¯à
<=> S=1+2S
<=> S=-1
©ú¥Ü§À¶µ¥iÁקKÅ]³Nºâªk°ÝÃD:
S=1+2+4+8+...+2^¡Û
<=> S=1+2(1+2+4+...+2^(¡Û-1))
<=> S=1+2S-2^(¡Û+1)
<=> S=2^(¡Û+1)-1 // ³oÃþ©¿²¤§À¶µ±o¨ìS=-1ªºÅ]³Nºâªk¨Ò¤l¦byoutube¤W¬Û·í¦h
// (§t¡Ûªº¥[¶µ³Q¥G²¤±¼)
©w²z1: s1=s2 <=> s1-s2=0
©w²z2: £U(n=0,¡Û) a(n)= a(0)+ £U(n=1,¡Û) a(n)
= a(¡Û)+ £U(n=0,¡Û-1) a(n)
©w²z3: £U(n=0,¡Û) f(n) ¡Ó £U(n=0,¡Û) g(n) = £U(n=0,¡Û) f(n)¡Óg(n)
©w²z4: £U(n=0,¡Û) c*f(n)= c*(£U(n=0,¡Û) f(n))
ÃÒ: ÃÒ²¤ (¥Ñ¯Å¼Æ®i¶}¦¡¥iÃÒ. ¤@¨Çº¾¸H³W«h¥¼¦C)
°ò¥»¤W,'¦³'¯Å¼Æªº¤½¦¡¥ç¾A¥Î©óµL¯Å¼Æ (¦ý,¼Æ¾ÇÂk¯Çªk¤£¾A¥Î©óµL¯Å¼Æ¤½¦¡ªº
ÃÒ©ú,¦]爲¡Ûªº¸q·N¬O'µ{§Ç¤£²×¤î',¦Ó©ó±À²z®É,Peano¤½²zªºÀ³¥Î¦¸¼Æ¥²¶·¦³).
µù: ³\¦h(¯S§O¦³Ãö£k,e)µL¯Å¼Æ¡¥µ¥¦¡¡¦¥i¥Ñ¥H¤W©w²zµý©ú¤£¦¨¥ß. ³oÃþ¦¡¤l¹ê»Ú¤W¬O
ªñ¦ü¦¡(·¥È).
¦p: £U(n=1,¡Û) 1/n² ¡Ü £k²/6
£U(n=0,¡Û) (-1^n)*(1/(2n+1)) ¡Ü £k/4
£U(n=0,¡Û) k^n/n! ¡Ü e^k
+------+
| ªþ¿ý |
+------+
ªþ¿ý1: ¶ÈÁ|Ó¨Ò¤l,²z½×¤Wªºℝ¥i¥H³o»ò½s(¼Æ¾Ç,¯u²z¬O½s¥X¨Óªº)¥H¤è«K¤½¦¡µý©ú:
Eℕ ::= {n| n爲Peano ¤½²z©Ò©w¸qªº¦ÛµM¼Æ(n∈ ℕ<0>), ¦ý¥]§tµL¦¸À³¥Î
"n∈ℕ => S(n)∈ℕ"©Ò±o¤§¼Æ(µL¦hÓµL¤j) }
Eℤ ::= {n| n∈Eℕ ©Î -n∈Eℕ }
ℝ ::= {p/q| p,q∈Eℤ, q>=2 }
ªþ¿ý2: ℝ,ℕ¶°¦X¶¡¤£¦s¦b1-1¹ïÀ³µ{§Ç.
ÃÒ: ³]X= {x| x爲Peano¤½³]©Ò©w¸qªº¼Æ¤Î¥]§tµ{§Ç¤£²×¤î©Ò¯àªíªº¼Æ(§YµLªø¼Æ¦r©Ò
¯àªí¹Fªº¦UºØ¼Æ}, «hX»Pℕªº¤¸¯À¶¡µLªk«Ø¥ß1-1¹ïÀ³ºâªk(³æ¯Â¦]爲²×¤î/¤£²×¤î
ì¦]). ¤S¦]X»Pℝ ¦Pºc, ¬Gℝ,ℕ¶°¦X¤£¥i1-1¹ïÀ³ (¦¹µý¦³ÂI°¨ªê,À³µL¤j°ÝÃD).
¦¹©Ê½è¶¡±µ»¡©ú,¦h¼Æℝ¤¤ªº¼Æ¬OµLªk¥H¦³¦r²Å»¡©úªº.
µù: ¥H¤WX¶°¦X¤]¥i¥Î©ó»¡©ú0.999...´`Àô¤p¼Æ¥»¨(·í§@1-¶i¨î¼Æ)´N¥i¥Î¨Ó©w¸q¤@
¦Pºcªº¹ê¼Æ¶°. µL½×¦p¦ó,"´`Àô¤p¼Æ0.999..."¨Æ¹ê¤W¥iªí¤@Ó«D±`¤jªº¼Æ¶°¦X.
ªþ¿ý3: §Ú¹ïµL¤j¡Ûªº¸gÅ礣¨¬. ³oùØ¥u´£¨Ñ¨Ç¸û½T©wªºì«h:
1.¥Ñ¹ê¼Æ©w¸q,µL¤j¡Û¤]¥]§t¦bℝ¤¤,µL¤j»P¤@¯ë¦³¼Æ¬Û¦P,¯à¹Bºâ,¯à¤ñ¤j¤p,¥u¬O¥¦
¦³µL¦hÓ. Á|¤@±`¨£¥Ù¬Þ¨Ò¤l: "1+¡Û =¡Û"»P"lim(x->¡Û) f(x)"ªºÁͪñ·§©À¥Ù¬Þ. ¦]Y
¬O,«h¤À¤£¥X¬O§_x+1,©Îx-1¨ºÓ¬OÁͪñ©Î»·Â÷¡Û. '¡Û'ªº»y·N¥²¶·°ß¤@.
µù:Y¹ê¼Æ¤£§tµL¤j,«h'¹L®Éªº¹ê¼Æ'¤£À³¦³µL¯Å¼Æ. '¹L®Éªº¹ê¼Æ'¤£¥i¯à¦P®Éµý©ú
爲¤@P»P§¹³Æ. Ä´¦p¤@¨Ç¥H¼Æ©w¸q¼Æªº¹ê¼Æ©w¸q(¤£¦¨¥ß) («Ü©êºp,¥»¤H¤£²M·¡¹ê»Ú
¤Wªº'¹L®Éªº¹ê¼Æ'¦³¦h¤ÖÓ. Ä´¦pªü°ò¦Ì¼w©Ê½èÀ³¬O¥Î©ó©w¸q¤@Ó¨S¦³µL¤j(©Î¤p
)ªº¼Æ¨t,¦ý¦¹¼Æ¨t«o¥i¨Ï¥ÎµL²Ö¥[¤Îlim(x->¡Û) ...? ¤SÄ´¦pDedekind cut²z½×,
°£¤F¤Wz°ÝÃD¥~,§ó«ÅºÙ"¥i«Ø³y¨C¤@¹ê¼Æ"...°²±Ôz). ¦]¦¹,Y¤@P(µL¥Ù¬Þ),«h
³Ì¦h,³oºØ¼Æ¨t¬O¦¹ÀÉ©w¸qªº¹ê¼Æªº¤l¶°.
Á`¦X¨Ó»¡,¹ê¼Æº¡¨¬ [1]¤T¤@«ß [2]¸Y±K(§t¨BÆJµL) [3]+*«Ê³¬(§t¨BÆJµL). ¦]¦¹,
¥i»¡¹ê¼Æ¶°ªº¯SìÝ´N¬O§tµL¤j¡Û.
2.µL¤j¡Ûªº»y·N°ò¥»¤W¬O«ü¤@Óµ{§Ç°j°é(¦p:Peano¤½²z). ¯S©Ê¬O"¥Ã¤£²×¤î" (¦¹©ó
µ{§Çªºµý©ú¤¤¥i¯à·N«ä¬O'¤£¥i¨M©w','¥¼©w¸q'©Î¡¥¥Ù¬Þ¡¥). ¥tÓ¬ÛÃö°ÝÃD¬O¸Y±K©Êªº
±Ôz¤£§¹¾ã(¦p¦PPeano¤½²z), ¦]¹ï©ó¦³²z¼Æ,¸Y±K©ÊªºÀ³¥Î¥²¶·²×¤î,¹ê¼Æ«hµL¦¹n¨D.
3.µL¤j¡Û¤]¥]§tCantor§Ç¼Æ,¦]爲'¼Æ'ªº·§©À«Üì©l. ÁöµM'§Ç¼Æ'¦³¨ä©w¸q,¥¦¤]¥Î¨ì¤F
¡@¡@p¼Æ·§©À(¥t,ℝ¤¤ªº¥[¼¹Bºâ¬O«Ê³¬ªº).
4.¦b¡¦¡Û¡¥»y·N°ß¤@ªº±¡ªp¤U,¥H¤U¤TºØe(¦ÛµM¹ï¼Æ©³)ªº'¼Æ'¸q·N¤£¦P:
e::= lim(n->¡Û) (1+1/n)^n = (1+1/¡Û)^¡Û
e^k= lim(n->¡Û) (1+k/n)^n = (1+k/¡Û)^¡Û
e^k= £U(n=0,¡Û) k^n/n!
5.³oÓ¨Ò¤lÀ³¦³§U©ó¸ÑÄÀµL¤j¡Ûªº·§©À:
³] A(0)=0
A(n)= (A(n-1)+1)/2
°Ý: A(¡Û)= ?
µª: 'A(¡Û)'Ȥ£¦b©Òµ¹¤©ªº¸Ñµª°ì¤¤. ¦¹µª®×¥ç¾A¥Î°ò¥»ªºZeno®¯½×,¶W¯Å¥ô°Èµ¥®¯½×.
ªþ¿ý4: ¤Gºû¼Æ¥iªí¥±. ¦b¤Gºû¼ÆªºÆ[ÂI¤U,¥unº¡¨¬¶ZÂ÷¤½³] (1.ÂI¸s¥²¾¤£§ïÅÜÂI¸s
¶¡ªº¶ZÂ÷ 2.ÂI¸sªº«Y¼Æ¿n¤£§ïÅÜÂI¸s¶¡ªº¶ZÂ÷¤ñ) §Y¥i«Øºc¼Ú¤ó´X¦ó¨t²Î.
³oùØ·Q»¡ªº¬O: ³oºØ'½èÂI¦t©z'¬O®Ú¾Ú§Ú̪º¹w³]©Ê½è©Ò«Øºcªº. §Ú̲רs¬O¦b±´¯Á
ª¾ÃѦۨªº»y·N. ¨Ã¥B,¥unÅ޿覨¥ß,À³³£¥i§ä¨ì¬ÛÀ³¨Æ¹ê. ¤Ï¦V¨Ó»¡,¥Ñª«²z¨Ó±´¯Á
'¹ê¼Æ'°ò¥»¤W¬O¦¨¥ßªº. ¨Ì¼Æ¦ì®É¥Nªº»¡ªk,¦t©z(»y·N¤W)¬OӤѵMªºpºâ¾÷.
ªþ¿ý5: ¦¬㪘(¦³·¥)ªºµL¯Å¼ÆÀ³¸Ó¨Ï¥Î"lim"©Î'¡Ü'(°£©w¸q¥~)¼g¦¨,Ä´¦p:
lim(k->¡Û) £U(n=0,k) (-1^n)*(1/(2n+1)) = £k/4 ©ÎªÌ
£U(n=0,¡Û) (-1^n)*(1/(2n+1))¡Ü £k/4 ³o様¸q·N¤ñ¸ûºë½TÂI,¸û¤£©ö¥Ç¿ù.
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