Pf ∑cn= converges→ lim c1=0, n→00 then for all the n existing M, satisfies ckM, if, l, then =g<1, and Ic=nl=< Mq the common radio of geometric progression 2Ma ∑ Mq"converge when[zk=1 Icn=" also converges
Pf: 1 1 0 1 1 1 1 1 converges lim 0, then for all the existing ,satisfies | | , | | if | | | |, then 1,and | | | | | | n n n n n n n n n n n n n n c z c z n M c z M z z z q z z c z c z Mq z → = = = = 0 0 1 0 the common radio of geometric progression 1 converge. when | | | | | |also converges. n n n n n n n Mq Mq z z c z = = = ，
2o,absolutely converge =0 If>cm=2 diverge, we can also get the conclusion with proof by contradiction In fact, there is one point z in=>E(=>zD) makins g that ∑ n Cn=3 Converge, 2c=2"converge, contradiction with the theorem hypothesis. =when(=)1=21 2cn=diverge
0 absolutely converge. n n n c z = 2 0 3 2 3 2 3 0 2 0 If diverge, we can also get the conclusion with proof by contradiction. In fact,there is one point in | | | | | | | | ,making that converge converge,contradiction wi n n n n n n n n n c z z z z z z c z c z = = = （ ） ， 2 0 th the theorem hypothesis. when | | | | diverge n n n z z c z = ,
3. Convergence circle and Convergence radius y B R R C B When a<B, the convergence domain is red and the divergence domain is blue
b Cb R C CR O x y When ,the convergence domain is red and the divergence domain is blue. b 3.Convergence Circle and Convergence radius
TH4,2.2 If c-pin> en+l are the coefficients, then the n-)o0 C n convergence radius of the power series is ,D≠0 R=1+∞,p=0 0,p=+ Pf Series 2cn m=according to d' Alembert theory, we can get n+1 lim n+1 =im n→)0 n→0
TH 4.2.2 If , are the coefficients, then the convergence radius of the power series is 1 lim n n n c c + → = 1 , n n c c + 1 , 0, , 0, 0, . R = + = = + Pf: 0 1 1 1 Series ,according to d'Alembert theory,we can get lim lim . n n n n n n n n n n n c z c z c z z c z c = + + + → → = =
() If p+0, when pl=<1, F<=series 2c,="converge n-=0 power series >cn="absolutely converge in circle =1 whenp=>1,=>-=power series 2c=diverge outside circle =l Using reduction to absurdity, supposing exists Zo outside the circle z Cn=o"converge, then exists z,,<=<=Theory 4.2.1 → series ∑ Cn= converge
0 0 0 1 (1).If 0, when 1, series converge 1 power series absolutely converge in circle ; 1 1 when 1, power series diverge outside circle ; Using reduction to absur n n n n n n n n n z z c z c z z z z c z z = = = = = 0 0 1 1 0 0 1 0 1 dity,supposing exists z outside the circle , 1 converge,then exists z , .Theory 4.2.1 series converge; n n n n n n z c z z z c z = = =