Important Math Series

1. {e^x} = 1 + x + \frac{{{x^2}}}{{2!}} +  \frac{{{x^3}}}{{3!}} + \cdots

2. \ln (1 + x) = x - \frac{{{x^2}}}{2} +  \frac{{{x^3}}}{3} + \cdots + {( - 1)^{n - 1}}\frac{{{x^n}}}{n} + \cdots

3. \ln (1 - x) =  - x - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - \cdots  - \frac{{{x^n}}}{n} - \cdots

4. {a^x} = 1 + \frac{{x\ln a}}{{1!}} + \frac{{{{(x\ln  a)}^2}}}{{2!}} + \frac{{{{(x\ln a)}^3}}}{{3!}} + \cdots

5. Sinx = x - \frac{{{x^3}}}{{3!}} +  \frac{{{x^5}}}{{5!}} - \cdots + \frac{{{{( - 1)}^{n - 1}}{x^{2n -  1}}}}{{(2n - 1)!}} + \cdots

6. Cosx = 1 - \frac{{{x^2}}}{{2!}} + \frac{{{x^4}}}{{4!}}  - \cdots  + {( - 1)^n}\frac{{{x^{2n}}}}{{(2n)!}} +  \cdots

7. Tanx = x + \frac{1}{3}{x^3} + \frac{2}{{15}}{x^5}  + \cdots

8. Secx = 1 + \frac{1}{2}{x^2} + \frac{5}{{4!}}{x^4} +  \frac{{16}}{{6!}}{x^6} + \cdots

9. {e^{Sinx}} = 1 + x + \frac{{{x^2}}}{2} -  \frac{{{x^4}}}{8} - \frac{{{x^5}}}{{15}} -  \cdots

10. Si{n^{ - 1}}x = \frac{x}{1} +  \frac{1}{2}\frac{{{x^3}}}{3} + \frac{{1 \cdot 3}}{{2 \cdot 4}}\frac{{{x^5}}}{5}  + \frac{{1 \cdot 3 \cdot 5}}{{2 \cdot 4 \cdot 6}}\frac{{{x^7}}}{7} + \cdots

11. {(1 + x)^{\frac{1}{x}}} = e(1 - \frac{1}{2}x +  \frac{{11}}{{24}}{x^2} + \cdots )

12. {(1 + x)^{ - \frac{1}{2}}} = 1 - \frac{1}{2}x +  \frac{{1 \cdot 2}}{{2 \cdot 4}}{x^2} - \frac{{1 \cdot 3 \cdot 5}}{{2 \cdot 4  \cdot 6}}{x^3} + \cdots

13. {(1 - x)^{ - \frac{1}{2}}} = 1 + \frac{1}{2}x +  \frac{{1 \cdot 2}}{{2 \cdot 4}}{x^2} + \frac{{1 \cdot 3 \cdot 5}}{{2 \cdot 4  \cdot 6}}{x^3} + \cdots

14. {(1 + x)^{\frac{1}{2}}} = 1 + \frac{1}{2}x -  \frac{1}{{2 \cdot 4}}{x^2} + \frac{{1 \cdot 3}}{{2 \cdot 4 \cdot 6}}{x^3}  + \cdots

15. \frac{1}{{1 - x}} = 1 + x + {x^2} + {x^3} + \cdots

16. \frac{1}{{1 + x}} = 1 - x + {x^2} - {x^3} + \cdots