收敛区域 收敛半径 幂级数的性质 Abel第二定理 幂级数的运算 Taylor 展开 §7.3幂级数与泰勒级数展开 本节我们将讨论一种简单的函数项级数“幂级数”： ∑anx”=a0+a1x+a2x2+…+anxn+…. n=0 幂级数的形式虽然简单，但它和后面要讨论的三角级数是应用最广泛也最重 要的两类函数项级数. 我们首先要研究清楚它的和函数的性质：定义域、连续性、可微性和可 积性. 返回 全屏 关闭 退出 1/17
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm §7.3 ?êV?êÐm !·ò?Ø«{ü¼ê?ê/?ê0 µ X ∞ n=0 anx n = a0 + a1x + a2x 2 + · · · + anx n + · · · . ?ê/ª,{ü, §Ú¡?Øn?ê´A^2 üa¼ê?ê. ·ÄkïÄÙ§Ú¼ê5µ½Â!ëY5!5Ú È5. 1/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm 7.3.1 ?êÂñ« ½n 1 XJ?ê3 x0 (6= 0) ?Âñ, K3¤k |x| |x0| uÑ. y² XJ?ê3 x0 6= 0 Âñ, Kéu?¿÷v |x| < |x0| x, k anx n = anx n 0 x x0 n Ï lim n→∞ anx n 0 = 0£ù´Ï?ê3 x0 Âñ¤, ¤± |anx n 0 | < M£k .¤, = |anx n | 6 M| x x0 | n , | x x0 | < 1, P∞ n=0 | x x0 | n Âñ¶Ïd P∞ n=0 |anx n| Âñ. y. 2/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ©Ûe, ?êÂñ E kn«Uµ 1 ◦ =3 x = 0 ?ÂñµE = {0}; 2 ◦ 3 E = (−∞, +∞) þ??Âñ; 3 ◦ kØ"Âñ:ÚuÑ:, ¤± E k. ½n 2 XJ?êk"Âñ:ÚuÑ:, Kkê R, ¦?ê3 (−R, R) ¥ýéÂñ, |x| > R , ?êuÑ. y² dukuÑ:, ¤± E ´k.8, E kþ(., P R. q duk"Âñ:, ¤± R > 0, âc¡©ÛÒ ½n(Ø. ¡þ¡ R ?êÂñ»¶(−R, R) ¡?êÂñ«m. y 3Ä ²x , ?êÂñ« E Ä þ´±:¥%«m, 3 ù«mSÜ, ?êØÂñ ýéÂñ. ´«mà:ÿØ(, I äN¯KäNé. 3/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm 7.3.2 Âñ»O ½n 3 XJ lim | an+1 an | = L; ½ lim sup pn |an| = L, K?ê P∞ n=0 anx n Âñ» R = 1 L = 0, L = +∞; 1 L , L k; +∞, L = 0. y² â D’Alembert O{l lim n→∞ an+1x n+1 anxn = lim n→∞ an+1 an |x| = L|x|. L|x| 1, = |x| > R , ?êuÑ. ¤±, ?êÂñ» R. 1ú ªâ Cauchy O{y². 4/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ~ 1 ¦?ê P∞ n=1 n αx n Âñ» R. ) Ï lim n α (n+1)α = lim n n+1α = 1, R = 1; α = 0µ P x n 3 x = ±1 uÑ, ¤± E = (−1, 1)£Ø¹à:¤. α = −1µ P 1 n x n 3 x = 1 uÑ, 3 x = −1 Âñ. ¤± E = [−1, 1)£¹ à:, عmà:¤. α = −2µ P 1 n2x n 3 x = ±1 ÑÂñ. ¤± E = [−1, 1]£¹mà :¤. þ¡~f`²3Âñ«mà:, «¹ÑUu). ~ 2 ¦?ê P∞ n=1 1 n! x n Âñ» R. ) Ï lim n! (n+1)! = lim 1 n+1 = 0, R = +∞. 5/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm 7.3.3 ?ê5 ?ê P∞ n=0 anx n 3 I = (−R, R) ¥Âñu S(x). ½n 4 ?ê3 I = (−R, R) S?Û4f«mþÂñ, Ï , Ú¼ê S(x) 3 I SëY. y² ? 0 < r < R, K P∞ n=0 |anr n|Âñ, |x| 6 r |anx n | 6 |anr n |, ¤± P∞ n=0 anx n 3 [−r, r] þÂñ. éu I ¥?¿4«m J, ½3 r, ¦ J ⊂ [−r, r] ⊂ (−R, R), ¤±3 J þÂñ. Ú¼êëY5K´ w,. 6/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ½n 5 ?êÚ¼ê S(x) 3Âñ«m I = (−R, R) ¥, ¿k S 0 (x) = X ∞ n=1 nanx n−1 , ¦?êÂñ»E R. y² k¦ P∞ n=1 nanx n−1 Âñ». ? x0 ∈ (−R, R), 3 r : |x0| R. XJ R0 > R, K3 x0µR0 > x0 > R, P |nanx n−1 0 | < ∞. Ï |anx n 0 | 6 |nanx n 0 | = x0|nanx n−1 0 | ¤± P |anx n 0 | Âñ, ù´ØU, ¤± R0 = R. 7/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ½n 6 ?êÚ¼ê S(x) 3Âñ«m I = (−R, R) SÈ, é x ∈ (−R, R) k Z x 0 S(t) dt = Z x 0 X ∞ n=0 ant n ! dt = X ∞ n=0 Z x 0 ant n dt = X ∞ n=0 an n + 1 x n+1 , ¿ È©?êÂñ»E R. ~ 3 ®?ê S(x) = P∞ n=0 x n n! 3ê¶þÂñ, K S 0 (x) = X ∞ n=1 x n−1 (n − 1)! = S(x) (−∞ < x < +∞), )d©§ S(x) = Aex . du S(0) = 1, S(x) = e x , = e x = X ∞ n=0 x n n! (−∞ < x < +∞). 8/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ~ 4 ¦?ê X ∞ n=0 nxn Ú. ) N´ù?êÂñ» 1, 3 x = ±1 ÑuÑ, Âñ «m (−1, 1). - S(x) = x P∞ n=1 nxn−1 , 2- f(x) = P∞ n=1 nxn−1 , 3«m [0, x] þÅÈ©, Z x 0 f(x)dx = X ∞ n=1 x n = x 1 − x , 2ò ªüàé x ¦ûÒ f(x) = 1 (1−x) 2 , ¤±?êÚ¼ê´ S(x) = x (1 − x) 2 , |x| < 1 ddq¦Ñ ê?êÚ. ~X©O- x = 1 2 Ú x = 1 3 , X ∞ n=1 n 2 n = 2, X ∞ n=1 n 3 n = 3 4 . 9/17 kJ Ik J I £ ¶ '4 òÑ
Âñ« Âñ» ?ê5 Abel 1½n ?ê$ Taylor Ðm ~ 5 ¦?ê X ∞ n=0 (−1)n 2n + 1 x 2n+1 Ú¼ê. ) Ï X ∞ n=0 (−1)nx 2n = X ∞ n=0 (−x 2 ) n = 1 1 + x2 , x ∈ (−1, 1), 3«m [0, x] þÅÈ© arctan x = X ∞ n=0 (−1)n 2n + 1 x 2n+1, x ∈ (−1, 1). 10/17 kJ Ik J I £ ¶ '4 òÑ