2ex=∞∑n=01n!xn= 1 + x +12!x2+ ··· +1n!xn+ ··· ,x ∈ (−∞,+∞)esinx= 1 + x +12x2−18x4−115x5−1240x6+190x7+315760x8+15670x9+ o(x9)etanx= 1 + x +12x2+12x3+38x4+37120x5+59240x6+137720x7+8715760x8+41641362880x9+ 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 ∈ (−∞,+∞)ln(1 + x) =∞∑n=0(−1)nn + 1xn+1= x −12x2+13x3− ··· +(−1)nn + 1xn+1+ ··· ,x ∈ (−1,1]lnÅ1 + x1 − xã=∞∑n=12x2n−12n − 1= 2x +23x3+25x5+27x7+29x9+ o(x9),x ∈ (−1,1)11 − x=∞∑n=0xn= 1 + x + x2+ x3+ ··· + xn+ ··· ,x ∈ (−1,1)(1 + x)12= 1 +12x −18x2+116x3−5128x4+7256x5−211024x6+332048x7−42932768x8+ o(x8),x ∈ (−1,1)(1 + x)−12= 1 −12x +38x2−516x3+35128x4−63256x5+2311024x6−4292048x7+643532768x8−1215565536x9+ o(x9),x ∈ (−1,1)(1 + x)13= 1 +13x −19x2+581x3−10243x4+22729x5−1546561x6+37419683x7−93559049x8+ o(x8),x ∈ (−1,1)(1 + x)−13= 1 −13x +29x2−1481x3+35243x4−91729x5+7286561x6−197619683x7+545359049x8−1358501594323x9+ o(x9),x ∈ (−1,1)(1 + x)32= 1 +32x +38x2−116x3+3128x4−3256x5+71024x6−92048x7+9932768x8−14365536x9+ o(x9),x ∈ (−1,1)(1 + x)−32= 1 −32x +158x2−3516x3+315128x4−693256x5+30031024x6−64352048x7+10939532768x8−23094565536x9+ o(x9),x ∈ (−1,1)(1 + x)−2= 1 − 2x + 3x2− 4x3+ 5x4− 6x5+ 7x6− 8x7+ 9x8− 10x9+ o(x9),x ∈ (−1,1)tanx =∞∑n=1B2n(−4)n(1 − 4n)(2n)!x2n−1= x +13x3+215x5+17315x7+ ··· +(−1)n−122n(22n− 1)B2n(2n)!x2n−1+ ···Çx2<π24åsecx =∞∑n=0(−1)nE2nx2n(2n)!= 1 +12x2+524x4+61720x6+2778064x8+ o(x8)arctanx =∞∑n=0(−1)n2n + 1x2n+1= x −13x3+15x5+ ··· +(−1)n2n + 1x2n+1+ ··· ,x ∈ [−1,1]arcsinx =∞∑n=0(2n)!4n(n!)2(2n + 1)x2n+1=x +16x3+340x5+5112x7+351152x9+ o(x9),x ∈ (−1,1)sinhx =∞∑n=0x2n+1(2n + 1)!= x +x33!+x55!+x77!+ ··· +x2n+1(2n + 1)!+ ···coshx =∞∑n=0x2n(2n)!= 1 +x22!+x44!+x66!+ ··· +x2n(2n)!+ ···tanhx =∞∑n=122n(22n− 1)B2nx2n−1(2n)!= x −13x3+215x5−17315x7+622835x9+ o(x9),|x| <π2sechx =∞∑n=0E2nx2n(2n)!= 1 −12x2+524x4−61720x6+138540320x8− ··· +E2n(2n)!x2n+ ··· ,(|x| <π2)arsinhx =∞∑n=0Ç(−1)n(2n)!22n(n!)2åx2n+1(2n + 1)= x −16x3+340x5−5112x7+351152x9+ o(x9),|x| < 1artanhx =∞∑n=0x2n+12n + 1= x +x33+x55+x77+ ··· +x2n+12n + 1+ ··· ,(|x| < 1)maikelaolin(常见麦克劳林公式)文字版1 3(1 + x)α= 1 +∞∑n=1α(α − 1)···(α − n + 1)n!xn= 1 + αx +α(α − 1)2!x2+ ··· +α(α − 1)...(α − n + 1)n!xn+ ··· ,x ∈ (−1,1)earcsinx= 1 + x +12x2+13x3+524x4+16x5+17144x6+13126x7+6298064x8+3254326x9+8177145152x10+ o(x10)earctanx= 1 + x +12x2−16x3−724x4+124x5+29144x6−11008x7−12198064x8−116372576x9+17321145152x10+ o(x10)eex= e + ex + ex2+5e6x3+5e8x4+13e30x5+203e720x6+877e5040x7+23e244x8+1007e17280x9+4639e145152x10+ o(x10)lnsinx = lnx −16x2−1180x4−12835x6−137800x8+ ··· + (−1)n22n−1B2nn(2n)!x2n+ ··· ,0 < x2< π2lncosx = −12x2−112x4−145x6−172520x8−3114175x10+ ··· + (−1)n22n−1(22n−1− 1)B2nn(2n)!x2n+ ··· ,x2<π24lntanx = lnx +13x2+790x4+622835x6+12718900x8+ ··· + (−1)n−122n(22n−1− 1)B2nn(2n)!x2n+ ··· ,0 < x2<π24cscx =∞∑n=0(−1)n+12(22n−1− 1)B2n(2n)!x2n−1=1x+16x +7360x3+3115120x5+127604800x7+ o(x7),x ∈ (0,π)cotx =∞∑n=0(−1)n22nB2n(2n)!x2n−1=1x−13x −145x3−2945x5−14725x7+ o(x7),x ∈ (0,π)cothx =∞∑n=022nB2n(2n)!x2n−1=1x+13x −145x3+2945x5− ··· +22nB2n(2n)!x2n−1− ··· ,(0 < |x| < π)cschx =∞∑n=02(22n−1− 1)B2n(2n)!x2n−1=1x−16x +7360x3−3115120x5+127604800x7+ o(x7),x ∈ (0,π)arcoshx = ln2x −∞∑n=1Ç(−1)n(2n)!22n(n!)2åx−2n2n= ln2x −Ç14x−2+332x−4+15288x−6+ ··· +Ç(−1)n(2n)!22n(n!)2åx−2n2n+ ···å,|x| > 1arccosx =π2−∞∑n=0(2n)!4n(n!)2(2n + 1)x2n+1=π2−ïx +16x3+340x5+5112x7+351152x9+ o(x9)ò,arccotx =π2−∞∑n=0(−1)n2n + 1x2n+1=π2−ñx −13x3+15x5+ ··· +(−1)n2n + 1x2n+1+ ···ô,(x2< 1)arcothx =1x+13x3+15x5+17x7+ ··· +1(2n + 1)x2n−1+ ··· ,x ∈ (|x| > 1)2 。