级数的基本性质 定理1若级数∑和∑v收敛,则级数∑(n,) =1 也收敛,且∑(an±vn)=∑"n±∑ 证设∑,∑v∑(,±vn)部分和分别为Sn,Tn,Hn, H=1 且令 lim s=a, limT=b n→ n→0 则Wn=(1±v)+(l2±v2)+…+(un土vn) =(1+l2+…+Ln)土(v1+V2+…+vn) S士T 所以 lime=lim(Sn±T)=a±b n→0 故∑(n±vn)=∑n土∑ 18 二. 级数的基本性质 1 1 n n 和 n n u v = = 也收敛, 且 1 ( ) n n n u v = 1 1 1 ( ) . n n n n n n n u v u v = = = = 定理1 若级数 收敛, 则级数 1 1 1 n n n n , , ( )的部分和 n n n u v u v = = = 1 1 2 2 ( ) ( ) ( ) W u v u v u v n n n = + + + = S T n n lim , lim n n n n S a T b → → = = 1 2 1 2 ( ) ( ) = + + + + + + u u u v v v n n 证 设 且令 分别为 则 所以 lim lim( ) n n n n n W S T a b → → = = 1 1 1 故 ( ) n n n n n n n u v u v = = = = , , , S T W n n n