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1、 2 ex= n=0 1 n!x n = 1 + x + 1 2!x 2 + + 1 n!x n + ,x (,+) esinx= 1 + x + 1 2x 2 1 8x 4 1 15x 5 1 240 x 6 + 1 90 x 7 + 31 5760 x 8 + 1 5670 x 9 + o (x9) etanx= 1 + x + 1 2x 2 + 1 2x 3 + 3 8x 4 + 37 120 x 5 + 59 240 x 6 + 137 720 x 7 + 871 5760 x 8 + 41641 362880 x 9 + o (x9) sinx = n=0 (1)n (2n + 1)
2、!x 2n+1 = x 1 3!x 3 + 1 5!x 5 + (1)n (2n + 1)!x 2n+1 + ,x (,+) cosx = n=0 (1)n (2n)! x2n= 1 1 2!x 2 + 1 4!x 4 + (1)n (2n)! x2n+ ,x (,+) ln(1 + x) = n=0 (1)n n + 1 xn+1= x 1 2x 2 + 1 3x 3 + (1)n n + 1 xn+1+ ,x (1,1 ln 1 + x 1 x = n=1 2x2n1 2n 1 = 2x + 2 3x 3 + 2 5x 5 + 2 7x 7 + 2 9x 9 + o(x9),x (1,1)
3、 1 1 x = n=0 xn= 1 + x + x2+ x3+ + xn+ ,x (1,1) (1 + x) 1 2= 1 + 1 2x 1 8x 2 + 1 16x 3 5 128x 4 + 7 256x 5 21 1024x 6 + 33 2048x 7 429 32768x 8 + o(x8),x (1,1) (1 + x) 1 2= 1 1 2x + 3 8x 2 5 16x 3 + 35 128x 4 63 256x 5 + 231 1024x 6 429 2048x 7 + 6435 32768x 8 12155 65536x 9 + o(x9),x (1,1) (1 + x)
4、1 3 = 1 + 1 3x 1 9x 2 + 5 81x 3 10 243x 4 + 22 729x 5 154 6561x 6 + 374 19683x 7 935 59049x 8 + o (x8) ,x (1,1) (1 + x) 1 3 = 1 1 3x + 2 9x 2 14 81x 3 + 35 243x 4 91 729x 5 + 728 6561x 6 1976 19683x 7 + 5453 59049x 8 135850 1594323x 9 + o (x9) ,x (1,1) (1 + x) 3 2 = 1 + 3 2x + 3 8x 2 1 16x 3 + 3 128
5、x 4 3 256x 5 + 7 1024x 6 9 2048x 7 + 99 32768x 8 143 65536x 9 + o (x9) ,x (1,1) (1 + x) 3 2= 1 3 2x + 15 8 x2 35 16x 3 + 315 128x 4 693 256x 5 + 3003 1024x 6 6435 2048x 7 + 109395 32768 x8 230945 65536 x9+ o(x9),x (1,1) (1 + x)2= 1 2x + 3x2 4x3+ 5x4 6x5+ 7x6 8x7+ 9x8 10 x9+ o(x9),x (1,1) tanx = n=1
6、B2n(4)n(1 4n) (2n)! x2n1= x + 1 3x 3 + 2 15x 5 + 17 315x 7 + + (1)n122n(22n 1)B2n (2n)! x2n1+ x2 2 4 secx = n=0 (1)nE2nx2n (2n)! = 1 + 1 2x 2 + 5 24x 4 + 61 720 x 6 + 277 8064x 8 + o(x8) arctanx = n=0 (1)n 2n + 1x 2n+1 = x 1 3x 3 + 1 5x 5 + + (1)n 2n + 1x 2n+1 + ,x 1,1 arcsinx = n=0 (2n)! 4n(n!)2(2n
7、 + 1) x2n+1=x + 1 6x 3 + 3 40 x 5 + 5 112x 7 + 35 1152x 9 + o(x9),x (1,1) sinhx = n=0 x2n+1 (2n + 1)! = x + x3 3! + x5 5! + x7 7! + + x2n+1 (2n + 1)! + coshx = n=0 x2n (2n)! = 1 + x2 2! + x4 4! + x6 6! + + x2n (2n)! + tanhx = n=1 22n(22n 1)B2nx2n1 (2n)! = x 1 3x 3 + 2 15x 5 17 315x 7 + 62 2835x 9 +
8、o(x9),|x| 2 sechx = n=0 E2nx2n (2n)! = 1 1 2x 2 + 5 24x 4 61 720 x 6 + 1385 40320 x 8 + E2n (2n)!x 2n + ,(|x| 2 ) arsinhx = n=0 (1)n(2n)! 22n(n!)2 x2n+1 (2n + 1) = x 1 6x 3 + 3 40 x 5 5 112x 7 + 35 1152x 9 + o(x9),|x| 1 artanhx = n=0 x2n+1 2n + 1 = x + x3 3 + x5 5 + x7 7 + + x2n+1 2n + 1 + ,(|x| 1)
9、3 (1 + x)= 1 + n=1 ( 1)( n + 1) n! xn= 1 + x + ( 1) 2! x2+ + ( 1).( n + 1) n! xn+ ,x (1,1) earcsinx= 1 + x + 1 2x 2 + 1 3x 3 + 5 24x 4 + 1 6x 5 + 17 144x 6 + 13 126x 7 + 629 8064x 8 + 325 4326x 9 + 8177 145152x 10 + o(x10) earctanx= 1 + x + 1 2x 2 1 6x 3 7 24x 4 + 1 24x 5 + 29 144x 6 1 1008x 7 1219
10、8064x 8 1163 72576x 9 + 17321 145152x 10 + o(x10) ee x = e + ex + ex2+ 5e 6 x3+ 5e 8 x4+ 13e 30 x5+ 203e 720 x6+ 877e 5040 x 7 + 23e 244x 8 + 1007e 17280 x 9 + 4639e 145152x 10 + o(x10) lnsinx = lnx 1 6x 2 1 180 x 4 1 2835x 6 1 37800 x 8 + + (1)n 22n1B2n n(2n)! x2n+ ,0 x2 2 lncosx = 1 2x 2 1 12x 4 1
11、 45x 6 17 2520 x 8 31 14175x 10 + + (1)n 22n1(22n1 1)B2n n(2n)! x2n+ ,x2 2 4 lntanx = lnx + 1 3x 2 + 7 90 x 4 + 62 2835x 6 + 127 18900 x 8 + + (1)n1 22n(22n1 1)B2n n(2n)! x2n+ ,0 x2 2 4 cscx = n=0 (1)n+12(22n1 1)B2n (2n)! x2n1= 1 x + 1 6x + 7 360 x 3 + 31 15120 x 5 + 127 604800 x 7 + o(x7),x (0,) co
12、tx = n=0 (1)n22nB2n (2n)! x2n1= 1 x 1 3x 1 45x 3 2 945x 5 1 4725x 7 + o(x7),x (0,) cothx = n=0 22nB2n (2n)! x2n1= 1 x + 1 3x 1 45x 3 + 2 945x 5 + 22nB2n (2n)! x2n1 ,(0 |x| 1 arccosx = 2 n=0 (2n)! 4n(n!)2(2n + 1) x2n+1= 2 x + 1 6x 3 + 3 40 x 5 + 5 112x 7 + 35 1152x 9 + o(x9) , arccotx = 2 n=0 (1)n 2n
13、 + 1x 2n+1 = 2 x 1 3x 3 + 1 5x 5 + + (1)n 2n + 1x 2n+1 + , (x2 1) sin(sinx) = x 1 3x 3 + 1 10 x 5 8 315x 7 + 13 2520 x 9 47 49896x 11 + o(x11) sin(tanx) = x + 1 6x 3 1 40 x 5 55 1008x 7 143 3456x 9 968167 39916800 x 11 + o(x11) sin(sinhx) = x 1 15x 5 1 90 x 7 + 1 5670 x 9 + 1 3150 x 11 + o(x11) sin(
14、arctanx) = x 1 2x 3 + 3 8x 5 5 16x 7 + 35 128x 9 63 256x 11 + o(x11) tan(tanx) = x + 2 3x 3 + 3 5x 5 + 181 315x 7 + 59 105x 9 + 3455 6237x 11 + o(x11) tan(sinx) = x + 1 6x 3 1 40 x 5 107 5040 x 7 73 24192x 9 + 41897 39916800 x 11 + o(x11) tan(arcsinx) = x + 1 2x 3 + 3 8x 5 + 5 16x 7 + 35 128x 9 + 63 256x 11 + o(x11)