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1、? ?10.1 1.?: (1)? P n=1 cosn n(n+1) ?. ?. | n+p P k=n+1 cosk k(k+1)| n+p P k=n+1 1 k(k+1) = n+p P k=n+1 (1 k 1 k+1) = 1 n+1 1 n+p+1 0,?N = 1 , ?n N?, | n+p P k=n+1 cosk k(k+1)| 1 n 0,? P n=1 an? P n=1 bn?,?N N,? ?n N?, | n+p P k=n+1 ak| , | n+p P k=n+1 bk| .?, n+p P k=n+1 ak n+p P k=n+1 uk n+p P k=n
2、+1 bk.?| n+p P k=n+1 uk| 0,?lim n S2n= lim n S2n+1= A,?N 0,? ?n N?, |S2n A| , |S2n+1 A| N?, |Sn A| 0,? P n=1 un?,?,?N,? ?n N?, n+p P k=n+1 un N?, (2n)u2n 2 2n P k=n+1 uk . (2n + 1)u2n+1 2n P k=n+1 uk+ 2n+1 P k=n+1 uk 1,? ?,?. (4) P n=1 1 n1+1/n. ?lim n 1 n1+1/n/ 1 n = 1,? P n=1 1 n ?,?. (5) P n=1 n2
3、 (31 n)n .?lim n n q n2 (31 n)n = 1 3, ?,?. (6) P n=2 n (lnn)n. ?lim n n q n (lnn)n = 0,?,?. (7) P n=1 1000n n! .?lim n 1000n+1 (n+1)! /1000 n n! = 0,?,?. (8) P n=1 (n!)2 (2n)!. ?lim n (n+1)!)2 (2n+2)! /(n!) 2 (2n)! = 1 4, ?,?. (9) P n=1 1 3n( n+1 n )n 2. ?lim n n q 1 3n( n+1 n )n 2 = e 3 0).?p 1?,?
4、R + 2 1 x(lnx)p = (lnx)1p 1p |+ 2 ?,? ?.?p = 1?,?R + 2 1 x(lnx)p = lnlnx|+ 2 ?,?. ?0 p 0).?lim n 1 n(lnlnn)q/ 1 n(lnn)1/2 =lim n (lnn)1/2 (lnlnn)q = lim x+ x1/2 (lnx)q = +.? P n=3 1 n(lnn)1/2 ?,?. 3.?:? P n=1 un?,? P n=1 u2 n ?.?,? ?. ?.? P n=1 un?,?lim n un= 0,?un?.?un M. ?u2 n Mun. ?,? P n=1 u2 n
5、?. ?,? P n=1( 1 n) 2 ?,? P n=1 1 n ?. 4.?:? P n=1 a2 n ? P n=1 b2 n ?,? P n=1 |anbn|, P n=1(a n+bn)2, P n=1 |an| n ?. ?.?,? P n=1(a 2 n+ b2n) ?.?|anbn| 1 2(a 2 n+ b2n), (an+ bn)2 3 2(a2 n+ b2n), ? P n=1 |anbn|, P n=1(a n+ bn)2 ?.?bn= 1 n, ? ? P n=1 |an| n ?. 5.?:? P n=1 un? P n=1 vn?,? (1) P n=1(u n
6、+ vn); (2) P n=1(u n vn); (3) P n=1 unvn. ?. (1)?,?un+ vn un,?,? P n=1(u n+ vn) ?. (2)?,?un= vn= 1 n ?,? P n=1(u n vn) ?0. (3)?,? ?un= vn= 1 n ?,? P n=1 unvn?. 6.?lim n nun= l,?0 l 0?n?un 0.?un 0.? lim n u2 n/ 1 n2 = l2,? P n=1 1 n2 ?,? P n=1 u2 n ?.?lim n un/ 1 n = l, ? P n=1 1 n ?,? P n=1 un?. ?10
7、.3 1.? (1) P n=1 (1)n1 (2n)2 .? P n=1 |(1) n1 (2n)2 | = P n=1 1 (2n)2 ?,?. (2) P n=1 (1)n+1 (2n1)p (p 0).? 1 (2n1)p ?0,?.? ? P n=1 1 (2n1)p ?p 1?,?p 1?,? ?. (3) P n=2 (1)n nlnn. ? 1 n lnn ?0,?.? P n=2 1 nlnn ? ?,?. (4) P n=1(1) n n1 n .?= P n=1(1) n 1 n P n=1(1) n 1 n, ? ?,?.?lim n n1 n / 1 n= 1, ?
8、P n=1 n1 n ?.? ?. (6) P n=1(1) n n! 3n2 .?lim n (n+1)! 3(n+1)2 / n! 3n2 = lim n n+1 32n+1 = 0.?, ?. 4 (7) P n=2(1) n 1 n sin n. ?lim n 1 n sin n/ 1 n2 = ,?. (8) P n=1(1) n+1tan n ( 2 0)? P n=1 n n+1un ?. ?.? 1 np ? n n+1 ?,? P n=1 un?,?,? ? P n=1 un np ? P n=1 n n+1un ?. 3.? P n=1 cosn np (0 1?,?0 1
9、?,? P n=1 1 np ?.?|cosn np | 1 np, ? P n=1 cosn np ? ?. (ii)?0 x0(x x0?,?nx0 x?.?, ? P n=1 an nx = P n=1 an nx0n x0 x ?. 6.? P n=1 un?,? P n=1 2n1 n un?. ?. |2n1 n un| 2|un|,? P n=1 |un|? P n=1 2n1 n |un|?. ?10.4 5 1.?. (1) P n=1(lnx) n. ?|lnx| 1 3 ?, | 1 xn sin 3n| 1 |x|n 3n, ?,?.? (,1 3) ( 1 3,+).
10、 2.?. (1) fn(x) = 1 2n+x2, x +. ?. lim n fn(x) = 0.?|fn(x) 0| 1 2n, ?lim n 1 2n = 0,? ?. (2) fn(x) = x4 + en, x +. ?.lim n fn(x) = x4 = x2.?|fn(x) x2| = en x4+en+x2 en en= e n 2, ?lim n e n 2= 0, ?. (3) fn(x) = ln(1 + x2 n2), (a) l x +l, (b) x +. ?. lim n fn(x) = 0. (a)?|fn(x) 0| x2 n2 l2 n2, ?lim n
11、 l2 n2 = 0,? ?(l,l)?. (b)?xn= n,?lim nf(xn) 0 = ln2 ? ?(,+)?. (4) fn(x) = n2x 1+n2x, 0 x 1. ?.lim n fn(x) = 1.?xn= 1 n2, ?lim nf(xn) 1 = 1 2 ? ?. 3.?. (1) P n=1(1) n n x2+n2, x +. ?. |(1)n n x2+n2| n n2 = 1 n3/2, ?M?,?. (2) P n=1( xn n xn+1 n+1 ), 1 x 1. ?. | P k=n+1 (x k k xk+1 k+1 )| = |x n+1 n+1
12、| 1 n, lim n 1 n = 0,?. (3) P n=1 sin nx 1+(x2+n2)3 , x +. ?. | sinnx 1+(x2+n2)3 | 1 n3, ?M?,?. (4) P n=1 x 1+4n4x2, x +. 6 ?. 1 + 4n4x2 4n2|x|,?| x 1+4n4x2| 1 4n2, ?M?,?. (5) P n=1 x2 (1+x)n, x +. ?. P k=n+1 x2 (1+x)k = x (1+x)n x 1+nx 1 n, lim n 1 n = 0,?. (6) P n=1 sinxsinnx n2+x2, 0 x 2. ?. 1 n2
13、+x2 1 n, lim n 1 n = 0,? 1 n2+x2 ?0.? 1 n2+x2 ?n?,?| n P k=1 sinx sinkx| = |cos x 2 cos x 2 cos(n + 1 2)x| 2, ?,?. (7) P n=1(1) n1x2enx2, x 0)?,?g(x)?(,+)?. ?.(i)?xn= 3n+1,?| P k=n+1 2ksin xn 3k| 2 n+1sin1 sin1, ? ?(,+)?. (ii)?x M,M?, |2nsin x 3n| |2 n x 3n| M( 2 3) n. ?M?,? ? P n=1 2nsin x 3n ?M,M?. (iii) |(2nsin x 3n) | = |(2 3) ncosx 3n| ( 2 3) n, ?M?,? P n=1(2 nsinx 3n) ? ?(,+)?.?g(x)?(,+)?. 6.?(x) = P n=1 1 nx ?1 + ,+)?( 0),? ? P n=1 lnn nx ?1+,+)?( 0),?(x)?(1,+
