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# Usual limited expansions in 0

## The limited developments resulting from the exponential

Let's start with the functions from the exponential : exponential, cosinus, sinus et cosinus hyperbolic and hyperbolic sine

 \begin{array}{rcl} e^x & = & \displaystyle \sum_{k=0}^n \dfrac{x^k}{k!}+ o\left( x^n\right)\\ \ cos(x) & = & \displaystyle \sum_{k=0}^n (-1)^k\dfrac{x^{2k}}{(2k)!}+ o\left( x^{2n}\ right)(\text{or } o(x^{2n+1}))\\ \sin(x) & = & \displaystyle \sum_{k=0}^n (-1)^k\dfrac{x ^{2k+1}}{(2k+1)!}+ o\left( x^{2n+1}\right)(\text{ou } o(x^{2n+2}))\\ \ text{ch}(x) & = & \displaystyle \sum_{k=0}^n \dfrac{x^{2k}}{(2k)!}+ o\left( x^{2n}\right)( \text{ or } o(x^{2n+1}))\\ \text{sh}(x) & = & \displaystyle \sum_{k=0}^n \dfrac{x^{2k+1} }{(2k+1)!}+ o\left( x^{2n+1}\right)(\text{ou } o(x^{2n+2}))\\ \end{array}

## The powers of 1 + x or 1 – x

Here are the limited expansions of the functions that are a power of 1+x or 1-x, such as the root or the inverse:

 \begin{array}{rcl} \alpha \in \mathbb{R},(1+x)^\alpha & = & 1 + \alpha x + \dfrac{\alpha(\alpha-1)}{2} x^2+ \ldots+ \\ &&\dfrac{\alpha(\alpha-1)\ldots (\alpha-(n-1))}{n!}x^n+o(x^n)\\ \ dfrac{1}{1-x} & = & \displaystyle \sum_{k=0}^nx^k+ o\left( x^{n}\right)\\ \dfrac{1}{1+x} & = & \displaystyle \sum_{k=0}^n (-1)^kx^k+ o\left( x^{n}\right)\\ \dfrac{1}{(1-x)^2} & = & \displaystyle \sum_{k=0}^n (k+1)x^k+ o\left( x^{n}\right)\\ \sqrt{1+x} & = & \displaystyle 1+\ dfrac{x}{2}- \dfrac{x^2}{8}+\ldots+\\ && (-1)^{n-1}\dfrac{1\times 3\times \ldots \times (2n- 3)}{2\times 4 \times \ldots \times (2n) } x^n+o\left( x^{n}\right)\\ \end{array}

## Limited development of the logarithm

Here is the formula for the 0-limited expansion of the logarithm.

 \begin{array}{rcl} \ln (1-x) & = & \displaystyle -\sum_{k=1}^n \dfrac{x^k}{k}+ o\left( x^{n} \right)\\ \ln (1+x) & = & \displaystyle \sum_{k=1}^n (-1)^{k-1}\dfrac{x^k}{k}+ o\left ( x^{n}\right)\\ \end{array}

## Limited developments of tan and tanh

Here are the expansions of the tangent, the cotangent and the hyperbolic tangent

 \begin{array}{rcl} \tan x & = & \displaystyle x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+ \dfrac{17}{315}x ^7+ o\left( x^{8}\right)\\ \\ \th x & = & \displaystyle x-\dfrac{1}{3}x^3+\dfrac{2}{15}x ^5- \dfrac{17}{315}x^7+ o\left( x^{8}\right)\\\end{array}

## Arcsin, Arccos, Arctan, Argch, Argsh, Argth

Here are the limited expansions of the reciprocal functions of cos, sin, tan, ch, sh, th

 \begin{array}{rcl} \arccos x & = & \displaystyle \dfrac{\pi}{2}-(x+\dfrac{x^3}{6}+\dfrac{3}{8}\dfrac{ x^5}{5}+\ldots + \\ &&\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac {x^{2n+1}}{2n+1} )+ o\left( x^{2n+2}\right)\\ \arcsin x & = & \displaystyle x+\dfrac{x^3}{6 }+\dfrac{3}{8}\dfrac{x^5}{5}+\ldots + \\ &&\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac{x^{2n+1}}{2n+1})+ o\left( x^{2n+2}\right)\\ \arctan x & = &\displaystyle \sum_{k=0}^n(-1)^k \dfrac{x^{2k+1}}{2k+1}+ o\left( x^{2n+2}\right)\ \ \text{argch } \left( \dfrac{1+x}{\sqrt{x}}\right) & = & \sqrt{2} - \dfrac{\sqrt 2}{\sqrt{12}}x +\dfrac{3\sqrt{2}}{160}x^2 +\dfrac{5\sqrt{2}}{896}x^3+\dfrac{35\sqrt{2}}{18432}x^ 4+o(x^4)\\ \text{argsh } x & = & \displaystyle x-\dfrac{x^3}{6}+\dfrac{3}{8}\dfrac{x^5}{ 5}+\ldots + \\ &&(-1)^n\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\ dfrac{x^{2n+1}}{2n+1})+ o\left( x^{2n+2}\right)\\ \tex t{argth } x & = &\displaystyle \sum_{k=0}^n \dfrac{x^{2k+1}}{2k+1}+ o\left( x^{2n+2}\right) \\ \end{array}

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