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Usual limited expansions in 0

The purpose of this article is to present the formulas of limited expansions, both usual and atypical. We will try to be exhaustive for this fact sheet

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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