# 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. $\frac{1}{{1 + x}} = 1 – x + {x^2} – {x^3} + \cdots$