常见麦克劳林公式.pdf

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1、常见麦克劳林公式∑1111xn2ne=x=1+x+x+


+x+


;x2(1;+1)n!2!n!n=0sinx121415161731819(9)e=1+x+xxxx+x+x+x+ox28152409057605670tanx12133437559613778718416419(9)e=1+x+x+x+x+x+x+x+x+x+ox2281202407205760362880∑1(1)n11(1)n2n+1352n+1sinx=x=xx+x


+x+


;x2(1;+1)(2n+1)!3!5!(2n+1)!n=

2、0∑1(1)n11(1)n2n242ncosx=x=1x+x


+x+


;x2(1;+1)(2n)!2!4!(2n)!n=0∑1(1)n11(1)nn+123n+1ln(1+x)=x=xx+x


+x+


;x2(1;1]n+123n+1n=0•1+x‹∑12x2n1222235799ln==2x+x+x+x+x+o(x);x2(1;1)1x2n13579n=11∑1n23n=x=1+x+x+x+


+x+


;x2(1;1)1xn=0111213547521633742988(1+x)2=

3、1+xx+xx+xx+xx+o(x);x2(1;1)2816128256102420483276811325335463523164297643581215599(1+x)2=1x+xx+xx+xx+xx+o(x);x2(1;1)2816128256102420483276865536111253104225154637479358(8)(1+x)3=1+xx+xx+xx+xx+ox;x2(1;1)3981243729656119683590491122143354915728619767545381

4、358509(9)(1+x)3=1x+xx+xx+xx+xx+ox;x2(1;1)3981243729656119683590491594323333213343576979981439(9)(1+x)2=1+x+xx+xx+xx+xx+ox;x2(1;1)281612825610242048327686553633152353315469353003664357109395823094599(1+x)2=1x+xx+xx+xx+xx+o(x);x2(1;1)281612825610242048327

5、68655362234567899(1+x)=12x+3x4x+5x6x+7x8x+9x10x+o(x);x2(1;1)‚Œ∑1B(4)n(14n)1217(1)n122n(22n1)B2tanx=2nx2n1=x+x3+x5+x7+


+2nx2n1+


x2<(2n)!315315(2n)!4n=1∑1(1)nEx2n1561277secx=2n=1+x2+x4+x6+x8+o(x8)(2n)!2247208064n=0∑1(1)n11(1)n2n+1352n+1arctanx=x=xx+x

6、+


+x+


;x2[1;1]2n+1352n+1n=0∑1(2n)!135352n+135799arcsinx=x=x+x+x+x+x+o(x);x2(1;1)4n(n!)2(2n+1)6401121152n=0∑1x2n+1x3x5x7x2n+1sinhx==x++++


++


(2n+1)!3!5!7!(2n+1)!n=0∑1x2nx2x4x6x2ncoshx==1++++


++


(2n)!2!4!6!(2n)!n=0∑122n(22n1)Bx2n1121762tanhx=2n=xx3+x5x7+

7、x9+o(x9);jxj<(2n)!31531528352n=1∑1Ex2n15611385Esechx=2n=1x2+x4x6+x8


+2nx2n+


;(jxj<)(2n)!22472040320(2n)!2n=0‚Œ∑1(1)n(2n)!x2n+11353535799arsinhx==xx+xx+x+o(x);jxj<122n(n!)2(2n+1)6401121152n=0∑1x2n+1x3x5x7x2n+1artanhx==x++++


++


;(jxj<1)2n+13572n+1n=0∑1(1)

8、
(n+1)(1)(1):::(n+1)(1+x)=1+xn=1+x+x2+


+xn+


;x2(1;1)n!2!n!n=1arcsinx1213541517613762983259817