常見麥克勞林公式大全

1、常見麥克勞林公式唐紹東1qq:10960848772016年5月18日114級天津大學(xué)仁愛學(xué)院金融工程2班n=0ex=11Xn=1+X+X2+Xn+,XE(X,+x)2!n!111esinx=1+X+X2X4X52815113etanx=1+X+X2+X3+X4+-228(1)n1sinX=X2n+1=X3!X3+5!X5+n=011311X6+X7+X8+X9+0(X9)2409057605670375913787141641X5+X6+X7+X8+X9+O(X9)12024072057603628801.(1)nX2n+1+，Xe(X，+x)11(-1)n=12!x2+4!x4+(2n”x

2、2n+,xe(x,+x)11(-1)nXn+1=X-X2+X3+Xn+1+,XE(1,123n+1In(=2n1=2x+3X3+2X5+2X7+9X9+O(X9),Xe(1,1)n=11=1-XXn=1+X+X2+X3+Xn+，XE(1,1)n=012X1.=1-X+=X2X3+28161151+XX2+X3398124372912143591丿33981243729“r33133八2816128256“、r31535315693丿22816128256(1+X)1=1+(1+x)-1(1+x)111572133429X2+X3X4+X5X6+X781612825610242048353563

3、231429X4X5+X6X7+128256102420481022154374X4+X5X6+X76561196837281976X665617X610243003X6X7+10242048815X7196839X7+2048643532768X8+o(x8),Xe(1,1)643512155X8X9+o(x9),XE(1,1)327686553693559X8+o(x8),Xe(1,1)5453135850+X8X9+O(X9),XE(1,1)59049159432399143X8X9+O(X9),XE(1,1)3276865536109395230945X8X9+o(x9),XE(1,1

4、)3276865536(1+X)2=1-2X+3X2-4X3+5X4-6X5+7X6-8X7+9X8-10X9+o(X9),XE(-1,1)丄尹B2n(4)n(14n)tanX=吋n=1住(1)nE2nX2nSeCX=(2n)!n=0嚴(yán)(1)narctanx=x2n+12n+1n=0arcsinX=(2n)!sinhX=1217X2n1=X+丁X3+X5+X7+31531515612771+2X2+24X4+720X6+8064X8+o(x8)=X1X3+1X5+X2n+1+352n+14n(n!)2(2n+1)13535X2n+1=X+6X3+40X5+112X7+1152n=0X2n+1X

5、3X5X7=X+3!+5!+7!+(2nW+X2n+1(2n+1)!n=0嚴(yán)X2nX2X4X6coshx=(2n!=1+2!+4!+!+麗+n=022n(22n1)BX2n1tanhx=2nn=1嚴(yán)EX2nsechX=(2nr=n=0嚴(yán)/(1)n(2n)!X2n+1arsinhx=22n(n!)2(2n+1)X2nX2n(2n)!(1)n122n(22n1)Bn2皿X2+(X2T),XE-1,1X9+o(X9),XE(-1,1)1217=XX3+X5X7+X9315315283562+o(x9),|x|211X2+5X4烏X6+385X8+?EX2n+，(|x|葺)22472040320(2n

6、)!2丿13535=XX3+40X5112X7+1152X9+o(x9)，|x|1n=0嚴(yán)X2n+1artanhx=2n+1X3X5=X+y+T+X7X2n+1t+2n+1+,(|X|1)n=0a(a1)(an+1)a(a1)a(a1).(an+1)(1+x)a=1+nxn=1+ax+2!x2+xn+,xn=115117136293258177earcsinx=1+x+x2+x3+x4+x5+x6+x7+x8+x9+x1+O(x1)324614412680644326145152丿1712911219116317321earctanx=1+x+x2x3x4+x5+x6x7x8x9+x10+0(

7、x10)624241441008806472576145152丿5e5e13e203e877e23e1007e4639e720504024417280145152丿122n1B37800 x8+(T)nnTx2n+,0 x2n23122n1(22n114175n(2n)!22n(22n11)Bn22nx2n+,0 x24n!n!E(-1,1)eex=e+ex+ex2+x3+x4+6811x2x46180lnsinx=lnx-1lncosx=-x22301x6283511x4x6x81245252017x2n+,x241762127皿anx=lnx+3x2+90 x4+2835x6+時x8+(1

8、)(-1)n+12(22n-11)B(2n)!cscx=n=0X(-1)n22nB2nx2n1(2n)!n=0Xcothx=n=012nx2n1=17x+x3+6360n(2n)!31127問x5+604800 x7+0(x7)，xE(0，n)cotx=21x5-9454725x7+o(x7),xE(0,n)22nB1(2n)!x+13x45x3+222nB945x5+x2n1，(0|x|113535x+6x3+40 x5+112x7+五x9+o(x9)1，1x-x3+x5+35x2n+1+2n+1,x21)sin(tanx)=x+-x3x5640sin(sinhx)=x吉x5擊x7+話命x9+竊x11+o(x1l)1353563=x3+8x5x7+128x92318159x+3x3+5x5+315x7+而x9+1110773x+x3x5x7x9+6405040241923535tan(arc