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The usual series expansions

The purpose of this article is to present the formulas of the expansions in integer series (DSE), usual as well as atypical. We will try to be exhaustive for this fact sheet

Expansions in whole series resulting from the exponential

Let's start with the functions from theexponential : exponential, cosinus, sinus and hyperbolic cosine and hyperbolic sine. Their radius of convergence is +∞ for each of them

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

The powers of 1 + x or 1 – x

Here are the expansions in whole series of the functions which are a power of 1+x or 1-x, such as the root or the inverse. Their radius of convergence is 1

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

Integer series expansion of the logarithm

Here is the formula for the entire series expansion of the logarithm. Its radius of convergence is equal to 1

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

Arcsin, Arccos, Arctan, Argch, Argsh, Argth

Here are the expansions in whole series of the reciprocal functions of cos, sin, tan, sh and th. The radius of convergence of these functions is 1.

 \begin{array}{rcl} \arccos x & = & \displaystyle \dfrac{\pi}{2}-\sum_{n=0}^{+\infty}\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n )}\dfrac{x^{2n+1}}{2n+1}\\ \arcsin x & = & \displaystyle \sum_{n=0}^{+\infty}\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n)}\dfrac{x^{2n+1}}{2n+1}\\ \arctan x & = &\ displaystyle \sum_{n=0}^{+\infty} (-1)^n \dfrac{x^{2n+1}}{2n+1}\\ \text{argsh } x & = & \displaystyle \ sum_{n=0}^{+\infty} (-1)^n\dfrac{1 \times 3 \times \ldots \times(2n-1)}{2 \times 2 \times \ldots \times(2n )}\dfrac{x^{2n+1}}{2n+1}\\ \text{argth } x & = &\displaystyle \sum_{n=0}^{+\infty} \dfrac{x^{ 2k+1}}{2k+1}\\\end{array}

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2 COMMENTS

  1. Hello, it seems to me that there is an error in the dev. in integer series of (1+x)^a, the summation index must start at n=1 if you take the term out at n=0 (1)
    Have a good day

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