Math Formulas
- $e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots \\$
- $a^x = 1 + x\,\ln a + \dfrac{(x\,\ln a)^2}{2!} + \dfrac{(x\,\ln a)^3}{3!} + \cdots$
- $\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \quad -1 < x \leq 1$
- $\ln(1+x) = \left(\dfrac{x-1}{x}\right) + \dfrac{1}{2}\left(\dfrac{x-1}{x}\right)^2+ \dfrac{1}{3}\left(\dfrac{x-1}{x}\right)^3 + \cdots \quad x \geq \dfrac{1}{2} $
- $\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \cdots$
- $\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + \cdots $
- $\tan x = x + \dfrac{x^3}{3} + \dfrac{2x^5}{15} + \cdots + \dfrac{2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!}
\quad -\dfrac{\pi}{2} < x < \dfrac{\pi}{2} $
- $\cot x = \dfrac{1}{x} - \dfrac{x}{3} - \dfrac{x^3}{45} - \cdots - \dfrac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi $
- $ \sec x = 1+ \dfrac{x^2}{2} + \dfrac{5x^4}{24} + \dfrac{61x^6}{720} + \cdots + \dfrac{E_n x^{2n}}{(2n)!}\quad -\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$
- $\csc x = \dfrac{1}{x} + \dfrac{x}{6} + \dfrac{7x^3}{360} + \cdots + \dfrac{2\left(2^{2n}-1\right)E_n x^{2n}}{(2n)!} \quad 0 < x < \pi $
- $ \sin^{-1} x = x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1 \cdot 3}{2 \cdot 4} \dfrac{x^5}{5} +\dfrac{1\cdot3\cdot5}{2\cdot4\cdot6}\dfrac{x^7}{7} + \c dots \quad -1 < x <1 $
- $\cos^{-1} x = \dfrac{\pi}{2} - \sin^{-1} x =
\dfrac{\pi}{2} - \left(x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1\cdot3}{2\cdot4}\dfrac{x^5}{5}+\cdots \right)
\quad -1 < x < 1$
- $ \tan^{-1} x = \left\{
\begin{aligned}
x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + \cdots & \quad -1 < x < 1 \\
\dfrac{\pi}{2} - \dfrac{1}{x} + \dfrac{1}{3x^3} - \dfrac{1}{5x^5} + \cdots & \quad x \geq 1 \\
-\dfrac{\pi}{2} - \dfrac{1}{x} + \dfrac{1}{3x^3} - \dfrac{1}{5x^5} + \cdots & \quad x < 1
\end{aligned} \right.$
- $\mathrm{cot^{-1}}\,x = \dfrac{\pi}{2} - \tan^{-1} x = \left\{
\begin{aligned}
\dfrac{\pi}{2} - \left(x - \dfrac{x^3}{3} + \dfrac{x^5}{5} + \cdots \right) & \quad -1 < x < 1 \\
\dfrac{1}{x} - \dfrac{1}{3x^3} + \dfrac{1}{5x^5} - \cdots & \quad x \geq 1 \\
\pi + \dfrac{1}{x} - \dfrac{1}{3x^3} + \dfrac{1}{5x^5} - \cdots & \quad x < 1
\end{aligned} \right.$