Contents of this article
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}Also find all our exercises limited development
Discover all our fact sheets:
- All properties of hyperbolic sines, cosines and tangents
- Common volume formulas
- The usual equivalents
- Form: Usual sums
- Form: All common primitives
- The usual perimeter formulas
- Trigonometry form: Sinus and cosine
- Multiplication tables
- Form: All usual derivatives
- The usual series expansions
