ex=1+x1!+x22!+...+xnn!e^x = 1 + \dfrac{x}{1!} + \dfrac{x^2}{2!} + ... + \dfrac{x^n}{n!}ex=1+1!x+2!x2+...+n!xn ax=1+xln(a)1!+(xln(a))22!+...(xln(a))nn!a^x = 1 + \dfrac{x\ln{(a)}}{1!} + \dfrac{{(x\ln(a)})^2}{2!} + ...\dfrac{(x\ln{(a)})^n}{n!}ax=1+1!xln(a)+2!(xln(a))2+...n!(xln(a))n sinx=x−x33!+...+(−1)nx2n+1(2n+1)!\sin{x} = x - \dfrac{x^3}{3!} + ... + \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}sinx=x−3!x3+...+(2n+1)!(−1)nx2n+1 cosx=1−x22!+...+(−1)nx2n(2n)!\cos{x} = 1 - \dfrac{x^2}{2!} + ... + \dfrac{(-1)^nx^{2n}}{(2n)!}cosx=1−2!x2+...+(2n)!(−1)nx2n
ex=1+x1!+x22!+...+xnn!e^x = 1 + \dfrac{x}{1!} + \dfrac{x^2}{2!} + ... + \dfrac{x^n}{n!}ex=1+1!x+2!x2+...+n!xn
ax=1+xln(a)1!+(xln(a))22!+...(xln(a))nn!a^x = 1 + \dfrac{x\ln{(a)}}{1!} + \dfrac{{(x\ln(a)})^2}{2!} + ...\dfrac{(x\ln{(a)})^n}{n!}ax=1+1!xln(a)+2!(xln(a))2+...n!(xln(a))n
sinx=x−x33!+...+(−1)nx2n+1(2n+1)!\sin{x} = x - \dfrac{x^3}{3!} + ... + \dfrac{(-1)^n x^{2n+1}}{(2n+1)!}sinx=x−3!x3+...+(2n+1)!(−1)nx2n+1
cosx=1−x22!+...+(−1)nx2n(2n)!\cos{x} = 1 - \dfrac{x^2}{2!} + ... + \dfrac{(-1)^nx^{2n}}{(2n)!}cosx=1−2!x2+...+(2n)!(−1)nx2n
1(1−x)=1+x+x2+...+xn\dfrac{1}{(1 - x)} = 1 + x + x^2 + ... + x^n(1−x)1=1+x+x2+...+xn 1(1+x)=1−x+x2−...+(−1)nxn\dfrac{1}{(1 + x)} = 1 - x + x^2 - ... + (-1)^nx^n(1+x)1=1−x+x2−...+(−1)nxn ln(1+x)=x−x22+x33−...+(−1)n+1xnn\ln{(1 + x)} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - ... + (-1)^{n+1}\dfrac{x^n}{n}ln(1+x)=x−2x2+3x3−...+(−1)n+1nxn ln(1−x)=−x−x22−x33−...−xnn\ln{(1 - x)} = -x - \dfrac{x^2}{2} - \dfrac{x^3}{3} - ... - \dfrac{x^n}{n}ln(1−x)=−x−2x2−3x3−...−nxn arctanx=x−x33+x55+...+(−1)nx2n+1(2n+1)\arctan{x} = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} +... + \dfrac{(-1)^n x^{2n+1}}{(2n+1) }arctanx=x−3x3+5x5+...+(2n+1)(−1)nx2n+1 (1+x)n=1+nx+n(n−1)x22!+n(n−1)(n−2)x33!+...(1 + x)^n = 1 + nx + \dfrac{n(n - 1)x^2}{2!} + \dfrac{n(n - 1)(n - 2)x^3}{3!} + ...(1+x)n=1+nx+2!n(n−1)x2+3!n(n−1)(n−2)x3+... 1(1−x)=1+(12)x−(38)x2+...\dfrac{1}{\sqrt{(1 - x)}} = 1 + \bigg(\dfrac{1}{2}\bigg)x - \bigg(\dfrac{3}{8}\bigg)x^2 +...(1−x)1=1+(21)x−(83)x2+...
1(1−x)=1+x+x2+...+xn\dfrac{1}{(1 - x)} = 1 + x + x^2 + ... + x^n(1−x)1=1+x+x2+...+xn
1(1+x)=1−x+x2−...+(−1)nxn\dfrac{1}{(1 + x)} = 1 - x + x^2 - ... + (-1)^nx^n(1+x)1=1−x+x2−...+(−1)nxn
ln(1+x)=x−x22+x33−...+(−1)n+1xnn\ln{(1 + x)} = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - ... + (-1)^{n+1}\dfrac{x^n}{n}ln(1+x)=x−2x2+3x3−...+(−1)n+1nxn
ln(1−x)=−x−x22−x33−...−xnn\ln{(1 - x)} = -x - \dfrac{x^2}{2} - \dfrac{x^3}{3} - ... - \dfrac{x^n}{n}ln(1−x)=−x−2x2−3x3−...−nxn
arctanx=x−x33+x55+...+(−1)nx2n+1(2n+1)\arctan{x} = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} +... + \dfrac{(-1)^n x^{2n+1}}{(2n+1) }arctanx=x−3x3+5x5+...+(2n+1)(−1)nx2n+1
(1+x)n=1+nx+n(n−1)x22!+n(n−1)(n−2)x33!+...(1 + x)^n = 1 + nx + \dfrac{n(n - 1)x^2}{2!} + \dfrac{n(n - 1)(n - 2)x^3}{3!} + ...(1+x)n=1+nx+2!n(n−1)x2+3!n(n−1)(n−2)x3+...
1(1−x)=1+(12)x−(38)x2+...\dfrac{1}{\sqrt{(1 - x)}} = 1 + \bigg(\dfrac{1}{2}\bigg)x - \bigg(\dfrac{3}{8}\bigg)x^2 +...(1−x)1=1+(21)x−(83)x2+...
par Nelly
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