? E=cf ?[) ? x1f ?? ￥Bá l ?à Q 1) ?/ f ??  ? U u×￥Bá l ?? (1) fn(x) = q x2 + 1n2;x 2 (?1;+1); (2) fn(x) = sin xn; i) x 2 (?l;l), ii) x 2 (?1;+1); (3) fn(x) = nx1+nx;x 2 (0;1); (4) fn(x) = 11+nx, i) x 2 [a;+1);a > 0; ii) x 2 (0;+1); (5) fn(x) = n2x 21+n3x3, i) x 2 [a;+1);a > 0; ii) x 2 (0;+1); (6) fn(x) = nx1+n+x;x 2 [0;1]; (7) fn(x) = xn1+xn, i) x 2 [0;b);b < 1; ii) x 2 [0;1], iii) x 2 [a;+1);a > 1; (8) fn(x) = xn ?x2n;x 2 [0;1]; (9) fn(x) = xn ?xn+1;x 2 [0;1]; (10) fn(x) = xn ln xn;x 2 (0;1); (11) fn(x) = 1n ln(1+e?nx);x 2 (?1;+1); 1 (12) fn(x) = e?(x?n)2; i)x 2 [?l;l] ii) x 2 (?1;+1): 2 !fn(x)(n = 1;2;￠￠￠)[a;b] μ?i Offn(x)g[a;b] Bá l ? p￡fn(x)[a;b] Báμ?. 3 !f(x)?l?(a;b) 7 fn(x) = [nf(x)]n (n = 1;2;￠￠￠): p￡ffn(x)g(a;b) Bá l ??f(x) . 4 !f(x)(a;b) =μ ??￥? ?f0(x) O fn(x) = n[f(x+ 1n)?f(x)]; p￡> uW[fi;fl](a < fi < fl < b) ffn(x)gBá l ??f0(x) . 5 !f1(x)[a;b]  ó ￡ V?lf ??  fn+1 = [nf(x)]n (n = 1;2;￠￠￠): p￡ffn(x)g[a;b] Bá l ?? ,. 6.ù? ?fi | I 1′ H fn(x) = nfixe?nx;n = 1;2;3￠￠￠ > uW[0;1] l ?$> uW[0;1]Bá l ?$ Plimn!1R10 fn(x)dx Vs |/ |K$ 7￡ ü? fn(x) = nxe?nx2;(n = 1;2;￠￠￠)> uW[0;1]  l ?? Z 1 0 limn!1fn(x)dx 6= limn!1 Z 1 0 fn(x)dx: 8 !ffn(x)(n = 1;2;￠￠￠)g(?1;+1)Bá ?? Offn(x)g (?1;+1)Bá l ??f(x) . p￡f(x)(?1;+1) Bá ??. 2 9 !ffn(x)g ^[a;b] ￥ ??f ?  Offn(x)g[a;b]Bá l ? ?f(x) ?xn 2 [a;b](n = 1;2;￠￠￠) ?@limn!1xn = x0 p￡limn!1fn(xn) = f(x0) 10 !ffn(x)g(a;b) =Bá l ??f(x);x0 2 (a;b) O limx?>x 0 fn(x) = an;(n = 1;2;￠￠￠): ￡ ülimn!1an?limx!x 0 f(x)i OM?' limn!1 limx!x 0 fn(x) = limx!x 0 limn!1fn(x) 11 !fn(x)(n = 1;2;￠￠￠)[a;b] ó ￡ V Offn(x)g[a;b]Bá l ? ?f(x)￡ üf(x)[a;b] ó ￡ V. x2f ?[) ?￥Bá l ??#  ?YE 1 p/ f ?[) ?￥ l ? u× '￥?Hq￥ (1) 1P n=1 xn 1+x2n; ó1P n=1 n n+1 ? x 2x+1 ￠n ; ?1P n=1 (?1)n 2n?1 ?1?x 1+x ￠n ; ?1P n=1 1p n ￠ 1 1+a2nx2: 2??l) ?/ f ?[) ?￥Bá l ?? (1) 1P n=0 (1?x)xn;x 2 [0;1]; ó1P n=1 (?1)n?1x2 (1+x2)n ;x 2 (?1;+1) . 3) ?/ f ?[) ?￥Bá l ?? 3 ò1P n=1 sinnx 3pn4+x4;x 2 (?1;+1); ó1P n=1 x 1+n4x2;x 2 (?1;+1) ?1P n=1 (?1)n(1?e?nx) n2+x2 ; x 2 [0;+1); ?1P n=1 sinx x+2n;x 2 (?2;+1); ?1P n=1 nx 1+n5x2;x 2 (?1;+1); ×1P n=1 n2p n!(x n +x?n); 12 ?jxj? 2; ?1P n=1 x2e?nx;x 2 (?1;+1); ù1P n=1 xn lnn x n! ;x 2 [0;1]; ú1P n=2 q x2 + 1n2 ? q x2 + 1(n?1)2 ? ; x 2 (?1;+1); ?1P n=1 n xn;jxj> r > 1; ü1P n=1 ln(1+nx) nxn ; x 2 [a;+1);a > 1: 4) ?/ f ?[) ?￥Bá l ?? (1) 1P n=1 cos 2n…3p n2+x2;x 2 (?1;+1); (2) 1P n=1 sinxsinnxp n+x ;x 2 [0;2…]; (3) 1P n=1 (?1)n x+n ;x 2 (?1;+1); (4) 1P n=1 (?1)n n+sinx;x 2 (?1;+1); 4 (5) 1P n=1 2n sin 13nx;x 2 (0;+1); (6) 1P n=1 (?1)n(n?1)2 3pn2+ex ;jxj6 a; (7) 1P n=1 xnp n;x 2 [?1;0]; (8) 1P n=1 (?1)nx2n+12n+1;x 2 [?1;1]. 5￡ ü) ?1P n=1 (?1)n?1 1n+x21?x(?1;+1) 1Bá l ?? ??xid ' l ? 7) ?1P n=1 x2 (1+x2)n ùx 2 (?1;+1)  ' l ? ?i?Bá l ?. 6 ! ?B[’n(x)? ^[a;b] ￥??f ? ?TP’n(x)[a;b]￥ ?1 ' l ? * 1?) ?[a;b] Bá l ?. 7 ?1P n=1 un(x)￥B?[jun(x)j? cn(x), x 2 X;i O1P n=1 cun(x)X B á l ?￡ ü1P n=1 un(x)X 9Bá l ? O ' l ?. x3???fff ? ? ?￥￥￥sss???ééé 1.ù?/ ) ? ?V U￥f ?·? uW ￥ ??? (1) 1P n=0 xn;?1 < x < 1; (2) 1P n=1 xn n ; ?1 ? x < 1; (3) 1P n=1 xn n2; jxj? 1; (4) 1P n=1 1 (x+n)(x+n+1); 0 < x < +1; (5) 1P n=1 1 1+n2x2; jxj > 0; 5 (6) 1P n=1 sinnx npn ; jxj < 1; (7) 1P n=1 nx 1+n4x2; jxj > 0; (8) 1P n=1 x2 (1+x2)n; jxj < 1: 2 p￡f(x) = 1P n=1 sinnx n3(?1;+1) = ??iμ ???f ?. 3 !f(x) = 1P n=1 e?nx 1+n2 p￡ òf(x)x > 0  ?? óf(x)x > 0 =í kQ V±. 4￡ ü1P n=1 ne?nx(0;+1) = ??. 5 !1P n=1 un(x)(a;b) =Bá l ?1P n=1 un(x)(n = 1;2;￠￠￠)[a;b]  ?? p￡ ò1P n=1 un(x)[a;b] Bá l ? óS(x) = 1P n=1 un(x)[a;b]  ??. 6 !) ?1P n=1 an(x) l ?￡ ü lim x?>0+ 1X n=1 an nx = 1X n=1 an: 7.￡ ü 1?r2 1?2rcosx+r2 = 1+2 1X n=1 an: 6 ?jr < 1j H? ?V7￡ ü Z … … 1?r2 1?2rcosx+r2dx = 2…(jrj < 1): 8.¨μK-?? ?￡ ü° D? ?. 9 !fxng ^(0;1) =￥B? ? '0 < xn < 1 Oxi 6= xj(i 6= j): k ) ?f ? f(x) = 1X n=1 sgn(x?xn) 2n (0;1)?￥ ???. 7