无穷积分柯西准则 第二积分中值定理 Dirichlet 判别法 Abel判别法 暇积分 第13章广义积分和含参变量积分 §13.1广义积分 13.1.1无穷区间上积分的收敛性 设 f(x) 在 [a,+∞) 的任何闭子区间上 Riemann 可积，则 f(x) 在无穷区 间[a,+∞)上积分的收敛性定义为如下极限 +∞ slenidie lim.sturdse 或者说定义为函数 F()=I furder. 的收敛性.类似数项级数，我们希望探讨广义积分收敛性的判别方法. 返回全屏关闭退出 1/36
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© 1 13 Ù 2ÂÈ©Ú¹ëCþÈ© §13.1 2ÂÈ© 13.1.1 á«mþÈ©Âñ5 f(x) 3 [a, +∞) ?Û4f«mþ Riemann È, K f(x) 3á« m [a, +∞) þÈ©Âñ5½ÂXe4 Z +∞ a f(x)dx = lim b→+∞ Z b a f(x)dx ½ö`½Â¼ê F(b) = Z b a f(x)dx, Âñ5. aqê?ê, ·F"&?2ÂÈ©Âñ5O{. 1/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© â¼ê4 lim b→+∞ F(b) Cauchy ÂñOK, 'u2ÂÈ© Cauchy ÂñOK. ½n 1 (Cauchy OK) f(x) 3 [a, +∞) ?Û4f«mþÈ, K R +∞ a f(x)dx Âñ¿©7^´éu?ê ε, 3 B > a, b1, b2 > B, Òk |F(b2) − F(b1)| = Z b2 b1 f(x)dx < ε. 5¿, X +∞ n=0 an Âñ =⇒ lim n→∞ an = 0 ´ Z +∞ a f(x)dx Âñ =6⇒ lim x→+∞ f(x) = 0. 2/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© ?Ú, a'?êÂñ5±, éuK¼ê, ·kXe(J: ½n 2 3 [a, +∞) þ f(x) > 0, f(x) 3 [a, +∞) ?Ûf«mþÈ, Káȩ R +∞ a f(x)dx Âñ¿©7^´, 3 M > 0 ¦é?Û b > a Ñk Z b a f(x)dx < M. = f(b) = R b a f(x)dx éu?Û b k. a'?êýéÂñÚ^Âñ, k ½n 3 f(x) 3 [a, +∞) ?Û4f«mþÈ, XJ R +∞ a |f(x)|dx  ñ, Káȩ R +∞ a f(x)dx Âñ. þ¡ù½nl Cauchy ÂñOK. 3/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© XJ Z +∞ a |f(x)|dx Âñ, K¡ Z +∞ a f(x)dx ýéÂñ. þ¡ù½n`² Z +∞ a f(x)dx 3á«mþýéÂñ¿XTÈ© Âñ. 5¿, 3éu~ÂÈ© (=k«mþÈ©), ù(ؿؤá, ~X f(x) = 1, x knê; −1, x Ãnê, T¼ê3 [0, 1] «mþýéÈ, ´ ØÈ. XJ Z +∞ a f(x)dx Âñ, Z +∞ a |f(x)|dx uÑ, K¡ Z +∞ a f(x)dx ^ Âñ. ~X, Z +∞ 0 sin x x dx ^Âñ. 4/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© aqá?ê'O{, k ½n 4 ('O{) f(x) Ú g(x) 3 [a, +∞) þk½Â, é?¿ b > a, f(x) Ú g(x) 3 [a, b] È. Xé¿© x, ¤áØ ª 0 6 f(x) 6 g(x), K 1 ◦ e Z +∞ a g(x)dx Âñ, K Z +∞ a f(x)dx Âñ; 2 ◦ e Z +∞ a f(x)dx uÑ, K Z +∞ a g(x)dx uÑ. 5/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© |^áȩ½Â±y² p > 1 Z +∞ a dx xp (a > 0) Âñ, p 6 1 Z +∞ a dx xp (a > 0) uÑ. éu3 [a, +∞) þëY¼ê f(x), ò§ 1 xp ', 1 ◦ eé¿© x, k |f(x)| 6 C xp , p > 1, C ~ê, K Z +∞ a f(x)dx ýéÂñ. 2 ◦ eé¿© x, k f(x) > C xp , p 6 1, C ~ê, Z +∞ a f(x)dx (a > 0) uÑ. 6/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© ½n 5 (Cauchy O{) XJ f(x) 3 [1, +∞) þk½ÂK üN~ ¼ê, @oÈ© R +∞ 1 f(x) dx ?ê P∞ n=1 f(n) ÓñÑ. y² d f(x) üN5, k 6 x 6 k + 1 k f(k + 1) 6 f(x) 6 f(k), u´ f(k + 1) 6 Z k+1 k f(x)dx 6 f(k). òþãØ ªé k = 1, 2, · · · , n \, Ò, é?Û n ∈ N k X n+1 k=2 f(k) 6 Z n+1 1 f(x)dx 6 X n k=1 f(k). e R +∞ 1 f(x)dx Âñ, Kdþª Pn+1 k=2 f(k) k., Ï P∞ n=1 f(n)  ñ. e R +∞ 1 f(x)dx uÑ, Kdþªm Pn k=1 f(k) Ã., P∞ n=1 f(n) uÑ. y. 7/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© ~ 1 y²3 [0, +∞) þ¼ê f(x) ¦ Z +∞ 0 f(x) dx Âñ, é?¿ê p 6= 1, Z +∞ 0 (f(x))p dx uÑ. y² e p 1, K- β = 2 − p. ok 0 < β < 1. ê an ∈ (0, 1 2 ), n = 1, 2, · · · , ¦ X ∞ k=1 ak = a < +∞, X ∞ k=1 a β k = +∞. ùê´3, X an = 1 n(ln(n + 4))2 . 8/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© E¼ê f(x) Xe: x ∈ [n − 1, n) , f(x) = an, n − 1 6 x 1 2 a p n + a 2−p n > 1 2 a β n . dd R +∞ 0 (f(x))p dx uÑ. 9/36 kJ Ik J I £ ¶ '4 òÑ
áȩ ÜOK 1È©¥½n Dirichlet O{ Abel O{ aÈ© ~ 2 p > 1, K Z +∞ 1 sin x xp dx Ú Z +∞ 1 cos x xp dx ýéÂñ. ~ 3 é?¿K¢ê α, Z +∞ 1 x αe −xdx Âñ. y² du lim x→+∞ x αe −x 2 = 0, x ¿©k x αe −x = x αe −x 2e −x 2 < e−x 2 . d Z +∞ 1 e −x 2dx Âñ5ÒíÈ©Âñ5. 10/36 kJ Ik J I £ ¶ '4 òÑ