e x= 1 + x +12! x 2+ ⋯ +1n! x n+ O(xn )\mathrm{e^x=1+x+\frac1{2!}x^2+ \cdots+\frac1{n!}x^n +O{(x^n)}}ex=1+x+2!1x2+⋯+n!1xn+O(xn)
l n ( 1 + x ) = x −1 2 x 2+1 3 x 3+ ⋯ + (−1)nn+1 xn+1+ O(xn )\mathrm{ln(1+x)=x-\frac12x^2+\frac13x^3+\cdots+\frac{(-1)^n}{n+1}x^{n+1}+O{(x^n)}}ln(1+x)=x−21x2+31x3+⋯+n+1(−1)nxn+1+O(xn)
( 1 + x)a = 1 + α x +α(α−1)2!x2 + ⋯ +α(α−1)⋯(α−n+1)n!xn+ O(xn )\mathrm{(1+x)^a=1+\alpha x+\frac{\alpha(\alpha-1)}{2!}x^2+\cdots+\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}x^n}+O{(x^n)}(1+x)a=1+αx+2!α(α−1)x2+⋯+n!α(α−1)⋯(α−n+1)xn+O(xn)
11−x = 1 + x +x 2+x 3+ ⋯ +x n+ O(xn )\mathrm{\frac1{1-x}~=1+x+x^2+x^3+\cdots +x^n + O{(x^n)} }1−x1 =1+x+x2+x3+⋯+xn+O(xn)
11+x= 1 − x +x 2−x 3+ ⋯ + ( − 1) n x n+ O(xn )\mathrm{\frac1{1+x}=1-x+x^2-x^3+\cdots +(-1)^\mathrm{n}\mathrm{x^n}+O{(x^n)} }1+x1=1−x+x2−x3+⋯+(−1)nxn+O(xn)
sin x = x −13! x 3+15! x 5+ ⋯ + (−1)n(2n+1)! x2n+1+ O (x2n+1) \mathrm{\sin x=x-\frac1{3!}x^3+\frac1{5!}x^5+\cdots+ \frac{(-1)^n}{(2n+1)!}x^{2n+1}+O(x^{2n+1})}sinx=x−3!1x3+5!1x5+⋯+(2n+1)!(−1)nx2n+1+O(x2n+1)
a r c s i n x = x +1 6 x 3+ ⋯ + (2n)!4n(n!)2(2n+1) x2n+1+ O (x2n+1) \mathrm{arcsin~x=x+\frac16x^3+\cdots+\frac{(2n)!}{4^n(n!)^2(2n+1)}x^{2n+1}+O(x^{2n+1})}arcsin x=x+61x3+⋯+4n(n!)2(2n+1)(2n)!x2n+1+O(x2n+1)
tan x = x +1 3 x 3+2 15 x 5+ ⋯ + B2 n ( − 4 ) n(1−4 n )(2n)! x2n−1+ O (x2n−1) \tan\mathrm{x}=\mathrm{x}+\frac13\mathrm{x}^3+\frac2{15}\mathrm{x}^5+ \cdots+\frac{\mathrm{B}_{2\mathrm{n}}\left(-4\right)^\mathrm{n}\left(1-4^\mathrm{n}\right)}{\left(2\mathrm{n}\right)!}\mathrm{x}^{2\mathrm{n}-1}+O(x^{2n-1})tanx=x+31x3+152x5+⋯+(2n)!B2n(−4)n(1−4n)x2n−1+O(x2n−1)
arctan x = x −13x3 +15x5 + ⋯ +(−1) n 2n+1x2n+1 + O (x2n+1 )\arctan\mathrm{x=x-\frac13x^3+\frac15x^5+\cdots+\frac{(-1)^n}{2n+1}x^{2n+1}+O(x^{2n+1})}arctanx=x−31x3+51x5+⋯+2n+1(−1)nx2n+1+O(x2n+1)
cos x = 1 −12!x2 +14!x4 + ⋯+ (−1)n(2n)! x2n+ O (x2n) \cos\mathrm{x=1-\frac1{2!}x^2+\frac1{4!}x^4+\cdots}+\frac{(-1)^n}{(2n)!}x^{2n}+O(x^{2n})cosx=1−2!1x2+4!1x4+⋯+(2n)!(−1)nx2n+O(x2n)