e x = 1 + x + 1 2 ! x 2 + ⋯ + 1 n ! x n + O ( x n ) \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 ) n n + 1 x n + 1 + O ( x n ) \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 ! x 2 + ⋯ + α ( α − 1 ) ⋯ ( α − n + 1 ) n ! x n + O ( x n ) \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)
1 1 − x = 1 + x + x 2 + x 3 + ⋯ + x n + O ( x n ) \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)
1 1 + x = 1 − x + x 2 − x 3 + ⋯ + ( − 1 ) n x n + O ( x n ) \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 − 1 3 ! x 3 + 1 5 ! x 5 + ⋯ + ( − 1 ) n ( 2 n + 1 ) ! x 2 n + 1 + O ( x 2 n + 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 + ⋯ + ( 2 n ) ! 4 n ( n ! ) 2 ( 2 n + 1 ) x 2 n + 1 + O ( x 2 n + 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 + ⋯ + B 2 n ( − 4 ) n ( 1 − 4 n ) ( 2 n ) ! x 2 n − 1 + O ( x 2 n − 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 − 1 3 x 3 + 1 5 x 5 + ⋯ + ( − 1 ) n 2 n + 1 x 2 n + 1 + O ( x 2 n + 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 − 1 2 ! x 2 + 1 4 ! x 4 + ⋯ + ( − 1 ) n ( 2 n ) ! x 2 n + O ( x 2 n ) \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)