Power series
We found 22 formulas for power series
A series of the form $$ \sum_{i=0}^\infty c_i x^i = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \ldots $$ is a power series at $x=0$.
7.
$\dfrac{1}{\sqrt{1+x}} = 1 - \dfrac{1}{2}x + \dfrac{1\cdot 3}{2\cdot 4}x^2 - \dfrac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1$
8.
$\sqrt{1+x} = 1 + \dfrac{1}{2}x - \dfrac{1}{2\cdot 4}x^2 + \dfrac{1\cdot 3}{2\cdot 4 \cdot 6}x^3 + \cdots \quad -1 < x \leq 1$
10.
$a^x = 1 + x\,\ln a + \dfrac{(x\,\ln a)^2}{2!} + \dfrac{(x\,\ln a)^3}{3!} + \cdots$
11.
$\ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + \cdots \quad -1 < x \leq 1$
12.
$\ln(1+x) = \left(\dfrac{x-1}{x}\right) + \frac{1}{2}\left(\dfrac{x-1}{x}\right)^2 + \dfrac{1}{3}\left(\dfrac{x-1}{x}\right)^3 + \cdots \quad x \geq \frac{1}{2}$
15.
$\tan x = x + \dfrac{x^3}{3} + \dfrac{2x^5}{15} + \cdots + \dfrac{2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$
16.
$\cot x = \dfrac{1}{x} - \dfrac{x}{3} - \dfrac{x^3}{45} - \cdots - \frac{2^{2n}B_nx^{2n-1}}{(2n)!} \quad 0 < x < \pi$
17.
$\arcsin x = x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1 \cdot 3}{2 \cdot 4} \frac{x^5}{5} + \dfrac{1\cdot3\cdot5}{2\cdot4\cdot6}\frac{x^7}{7} + \cdots \quad -1 < x <1$
18.
$\arccos x = \dfrac{\pi}{2} - \arcsin x = \dfrac{\pi}{2} - \left(x + \dfrac{1}{2}\dfrac{x^3}{3} + \dfrac{1\cdot3}{2\cdot4}\frac{x^5}{5}+\cdots \right) \quad -1 < x < 1$
21.
$\tanh x = x - \dfrac{x^3}{3} + \dfrac{2x^5}{15} + \cdots \dfrac{(-1)^{n-1} 2^{2n}\left(2^{2n}-1\right)B_nx^{2n-1}}{(2n)!} + \cdots \quad |x| < \frac{\pi}{2}$
22.
$\coth x = \dfrac{1}{x} + \dfrac{x}{3} - \dfrac{x^3}{45} + \cdots \dfrac{(-1)^{n-1} 2^{2n}B_nx^{2n-1}}{(2n)!} + \cdots \quad 0 < |x| < \pi$