王 例2求下列幂级数的收敛区间: 上()∑(-1y;:(2)∑(-mxy; -=1 n n=1 (3)∑ 2 (4)∑(-1”=n(x-2) n=1 n= 中解(1ae=n+1 =1∴R=1 n→0 n→ 生当=时,级数为,该级数收敛 = 庄当=时,级数为∑1该级数发散 nE 上或
例2 求下列幂级数的收敛区间: 解 (1) n n n a a 1 lim + → = 1 lim + = → n n n = 1 R = 1 当x = 1时, 当x = −1时, , ( 1) 1 = − n n n 级数为 , 1 1 n= n 级数为 该级数收敛 该级数发散 (1) ( 1) ; 1 n x n n n = − (2) ( ) ; 1 = − n n nx ; ! (3) 1 n= n n x ) . 2 1 ( 2 (4) ( 1) 1 n n n n x n − − =
故收敛区间是(-1,1 (2)∑(-nx)y; n=1 p= lima,=limn=+∞,∴:R=0, n→>00 n→>00 级数只在x=0处收敛, n (3)∑ n-=1 p=lim n+1=lim 1 =0,∴R=+ n→ n n→0n+1 收敛区间(∞,+0) 上或
故收敛区间是(−1,1]. n n n a → = lim n n→ = lim = +, R = +, 级数只在x = 0处收敛, n n n a a 1 lim + → = 1 1 lim + = n→ n = 0, R = 0, 收敛区间(−,+). (2) ( ) ; 1 = − n n nx; ! (3) 1 n= n n x