高等数学《概率论与数理统计》常见麦克劳林公式汇总.pdf

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1、ex=n=01n!xn=1+x+12!x2+1n!xn+,x (,+)esinx=1+x+12x218x4115x51240 x6+190 x7+315760 x8+15670 x9+o(x9)etanx=1+x+12x2+12x3+38x4+37120 x5+59240 x6+137720 x7+8715760 x8+41641362880 x9+o(x9)sinx=n=0(1)n(2n+1)!x2n+1=x 13!x3+15!x5 +(1)n(2n+1)!x2n+1+,x (,+)cosx=n=0(1)n(2n)!x2n=1 12!x2+14!x4 +(1)n(2n)!x2n+,x (,+

2、)ln(1+x)=n=0(1)nn+1xn+1=x 12x2+13x3 +(1)nn+1xn+1+,x (1,1ln1+x1 x=n=12x2n12n 1=2x+23x3+25x5+27x7+29x9+o(x9),x (1,1)11 x=n=0 xn=1+x+x2+x3+xn+,x (1,1)(1+x)12=1+12x 18x2+116x35128x4+7256x5211024x6+332048x742932768x8+o(x8),x (1,1)(1+x)12=1 12x+38x2516x3+35128x463256x5+2311024x64292048x7+643532768x8121556

3、5536x9+o(x9),x (1,1)(1+x)13=1+13x 19x2+581x310243x4+22729x51546561x6+37419683x793559049x8+o(x8),x (1,1)(1+x)13=1 13x+29x21481x3+35243x491729x5+7286561x6197619683x7+545359049x81358501594323x9+o(x9),x (1,1)(1+x)32=1+32x+38x2116x3+3128x43256x5+71024x692048x7+9932768x814365536x9+o(x9),x (1,1)(1+x)32=1 3

4、2x+158x23516x3+315128x4693256x5+30031024x664352048x7+10939532768x823094565536x9+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=1B2n(4)n(1 4n)(2n)!x2n1=x+13x3+215x5+17315x7+(1)n122n(22n 1)B2n(2n)!x2n1+x224secx=n=0(1)nE2nx2n(2n)!=1+12x2+524x4+61720 x6+2778064x8+o(x8)arc

5、tanx=n=0(1)n2n+1x2n+1=x 13x3+15x5+(1)n2n+1x2n+1+,x 1,1arcsinx=n=0(2n)!4n(n!)2(2n+1)x2n+1=x+16x3+340 x5+5112x7+351152x9+o(x9),x (1,1)sinhx=n=0 x2n+1(2n+1)!=x+x33!+x55!+x77!+x2n+1(2n+1)!+coshx=n=0 x2n(2n)!=1+x22!+x44!+x66!+x2n(2n)!+tanhx=n=122n(22n 1)B2nx2n1(2n)!=x 13x3+215x517315x7+622835x9+o(x9),|x|

6、2sechx=n=0E2nx2n(2n)!=1 12x2+524x461720 x6+138540320 x8 +E2n(2n)!x2n+,(|x|2)arsinhx=n=0(1)n(2n)!22n(n!)2x2n+1(2n+1)=x 16x3+340 x55112x7+351152x9+o(x9),|x|1artanhx=n=0 x2n+12n+1=x+x33+x55+x77+x2n+12n+1+,(|x|1)常见麦克劳林公式1(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+12x2+13x3+52

7、4x4+16x5+17144x6+13126x7+6298064x8+3254326x9+8177145152x10+o(x10)earctanx=1+x+12x216x3724x4+124x5+29144x611008x712198064x8116372576x9+17321145152x10+o(x10)eex=e+ex+ex2+5e6x3+5e8x4+13e30 x5+203e720 x6+877e5040 x7+23e244x8+1007e17280 x9+4639e145152x10+o(x10)lnsinx=lnx 16x21180 x412835x6137800 x8+(1)n2

8、2n1B2nn(2n)!x2n+,0 x2 2lncosx=12x2112x4145x6172520 x83114175x10+(1)n22n1(22n1 1)B2nn(2n)!x2n+,x224lntanx=lnx+13x2+790 x4+622835x6+12718900 x8+(1)n122n(22n1 1)B2nn(2n)!x2n+,0 x224cscx=n=0(1)n+12(22n1 1)B2n(2n)!x2n1=1x+16x+7360 x3+3115120 x5+127604800 x7+o(x7),x (0,)cotx=n=0(1)n22nB2n(2n)!x2n1=1x13x 145x32945x514725x7+o(x7),x (0,)cothx=n=022nB2n(2n)!x2n1=1x+13x 145x3+2945x5 +22nB2n(2n)!x2n1 ,(0|x|1arccosx=2n=0(2n)!4n(n!)2(2n+1)x2n+1=2x+16x3+340 x5+5112x7+351152x9+o(x9),arccotx=2n=0(1)n2n+1x2n+1=2x 13x3+15x5+(1)n2n+1x2n+1+,(x2 1)2