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1、?3(?)1.3.1.?Pan(x + 3)n?x = 5?, x = 0?,?x = 1?,?.?:?,?Pan(x+3)n?x = 5?Panxn?x = 2?,?x?|x| 2?Panxn?.?Pan(x + 3)n?x = 0?,?Panxn?x = 3?,?x?|x| 1,?R = 1.?|x| = 1?P 12n,?,?-1,1.(6)?un=3n+ (2)n n,?lim nnp |un| = 3,?R =1 3.?2x + 1 = 13,?x = 23?,?P 3n+ (2)n n1 3n(1)n= P?(1)n n+1 n (23)n?,?;?2x + 1 =1 3,?x =
2、 13?,?P 3n+ (2)n n1 3n=P?1 n+(1)n n (23)n?,?.?23,1 3).1(8)?lim n?an+1an? = limn(n n + 1)n=1 e,?R2=e,?R =e.?x = e?Pn!?e n?n,?0?,?.?(e,e).1.3.3.?Maclaurin?:(2) sin3x;(6) (1+x)ex;(7)x 1 + x 2x2;(8) ln(x+1 + x2);(10)Zx0cost2dt.?(2)?sinx =Pn=1(1)n+1x2n1 (2n 1)!,x (,),?sin3x =1 4(3sinx sin3x)=3 4Xn=1(1)n+
3、1x2n1 (2n 1)!1 4Xn=1(1)n+1(3x)2n1 (2n 1)!=1 4Xn=2(1)n(32n1 3) (2n 1)!x2n1,x (,).(6)?ex=Pn=0xn n!,?ex=Pn=0(1)nxn n!.?(1+x)ex=Xn=0(1)nxn(1 + x) n!= 1+Xn=1 or 2(1)n1h1 (n 1)!1 n!i xn,|x| .(7)?x 1 + x 2x2=1 3?1 1 x1 1 + 2x?1 1 x=Xn=0xn,|x| 1,1 1 + 2x=Xn=0(1)n(2x)n,|x| 1 2,?x 1 + x 2x2=Xn=01 (2)n 3xn, |x
4、| 1 2.(8)?11 + t2= 1 +Xn=1(1)n(2n 1)! (2n)!t2n, t 1,1,?ln(x +1 + x2) =Zx011 + t2dt=Zx0h 1 +Xn=1(1)n(2n 1)! (2n)!t2ni dt= x +Xn=1(1)n(2n 1)! (2n)!(2n + 1)x2n+1,x 1,1.2(10)?cost2=Pn=0(1)n(t2)2n (2n)!=Xn=0(1)nt4n (2n)!,t (,),?Zx0cost2dt =Zx0Xn=0(1)nt4n (2n)!dt =Xn=0(1)nx4n+1 (2n)!(4n + 1),x (,).1.3.6.?
5、:(2)Xn=1(1)nn2xn;(4)Xn=1(2n + 1)xn;(6)Xn=1(1)n1x2n (2n 1)32n1.?(2)?1.3.14?Xn=1n2xn=x(1 + x) (1 x)3, |x| 1,?Xn=1(1)nn2xn=x(x 1) (1 + x)3, |x| 1.(4) S(x) = 2Xn=1nxn+Xn=1xn=2x (1 x)2+x 1 x=x(3 x) (1 x)2, |x| 1.?:?S(x) =Pn=1(2n + 1)xn.?Pn=1(2n + 1)xn=Xn=12(n + 1)xnXn=1xn,?g(x) =Pn=12(n + 1)xn,h(x) =Pn=1
6、xn,?g(x)?Zx0g(x)dx =2Xn=1xn+1=2x2 1 x,?g(x) =2x(2 x) (1 x)2.?h(x) =Pn=1xn=x 1 x,?S(x) =Pn=1(2n + 1)xn=x(3 x) (1 x)2.(6)?S(x) =Xn=1(1)n1x2n (2n 1)32n1= xXn=1(1)n1x2n1 (2n 1)32n1= xg(x),?xg(x) =Xn=1(1)n1x2n1 32n1=x 3Xn=1? x2 9?n1 =x 31 1 +x2 9=3x 9 + x2.?g(x) =3 9 + x2,?g(x) =Rx 0g(x)dx + g(0) = arcta
7、nx 3.?S(x) = xarctanx 3, |x| 3.1.3.7.?(R,R)?f(x) =Pn=0anxn.?:?f?,?a2n= 0;?f?,?a2n+1= 0,?n N.?f(x) =Pn=0anxn,x (R,R),?f(x) =Pn=0(1)nanxn.?f?,?an+ (1)nan= 0(n = 1,2,3.).?n = 2k 1(k =31,2,.)?,?1 + (1)n= 0,?a2n= 0.?f?,?an (1)nan= 0(n = 1,2,3.).?n = 2k(k = 1,2,.)?,?1 (1)n= 0,?a2n+1= 0.1.3.8.?: (2)Pn=0(1)
8、n(n2 n + 1)2n.?(2)?S(x) =Pn=0(1)n(n2 n + 1)xn,?S(x) =Pn=0(1)nn2xnPn=0(1)nnxn+Pn=0(1)nxn=x(x 1) (1 + x)3x (1 + x)2+1 1 + x=x2+ 1 (1 + x)3.?x =1 2?,?Pn=0(1)n(n2 n + 1)2n,?Pn=0(1)n(n2 n + 1)2n=10 27.1.3.9.?f(x) =Pn=1n3n1xn1.(1)?f(x)?(1/3,1/3)?;(2)?Z1/80f(x)dx.?(1)?lim nnn3n1= 3,?1 3.?(1/3,1/3)?,?f(x)?(
9、1/3,1/3)?(2)?0,1 8?,?Z1/80f(x)dx =Xn=1Z1 80nxn13n1dx =1 5.1.3.13.?C()(1 + x)?x = 0?x2010?,I =Z10C(y 1)?1 y + 1+1 y + 2+1 y + 3+ +1 y + 2010? dy.?C(y 1) =(y 1)(y 2)(y 2010) 2010!=(y + 1)(y + 2)(y + 2010) 2010!,?d dy?(y + 1)(y + 2)(y + 2010)2010!? .?,I =(y + 1)(y + 2)(y + 2010) 2010!?10= 2011 1 = 2010
10、.41.3.17.?: (1)?Pn=0an?,?f(x) =Pn=0anxn?0,1?; (2)?Pn=0an?S,?lim x1Pn=0anxn= S.?(1)?|xn| 1 (x 0,1)?x 0,1, xn?n?,?Abel?Pn=0anxn?0,1?.(2)?.1.3.19.?Pan?, Sn?,?lim nan an+1?.?Sn+,an/Sn 0 (n ),?Panxn?. (?:?Stolz?.)?Sn ,?limnan an+1?,?Stolz?limnan Sn=limnan an1 Sn Sn1= limn(1 an1 an) = 0.?limnan1 an= 1,?Pa
11、nxn?1.1.3.20. ?:S = 1 1 4+1 71 10+ +(1)n+1 3n 2+ .?an=(1)n+1 3n 2= (1)n+1Z10x3n3dx,?Sn=Z10(1 x3+ x6 + (1)n+1x3n3)dx=Z101 (1)nx3n 1 + x3dx =Z101 1 + x3dx Z10(1)nx3n 1 + x3dx.?Z10(1)nx3n 1 + x3? 1 3n + 1 0 (n ),?S =Z101 1 + x3dx =1 3Z101 1 + xdx +Z102 x 1 x + x2dx=1 3ln(1 + x) 1 2ln(1 x + x2) +3arctan2x 13 |1t=0=1 3(ln2 +33).5
