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Series Expansion

Tool to calculate series expansions (Taylor, etc.) allowing an approximation of a mathematical function or expression.

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Series Expansion -

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# Series Expansion

## Series Expansion Calculator

### What is a series expansion? (Definition)

In mathematics, a series expansion of a function in the vicinity of a defined point is a polynomial expression allowing an approximation of this function. The limited expansion is therefore composed of a polynomial function (sum of polynomials) and a remainder which is small (or negligible) around the point.

### How to calculate a series expansion?

To compute a (limited) series expansion of order $n$ of a function $f(x)$ in the neighborhood of a value $a$, if the function is differentiable in $a$, then it is possible to use the Taylor-Young formula which decomposes any function into:

$$f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f^{(2)}(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^{n} + O(x^{n+1}) \\ = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^{k} + O(x^{n+1})$$

with $O(x^n)$ the Big O (Landau's asymptotic notation) indicating precision, a value tending to be negligible with respect to $(x-a)^n$ in the neighborhood of $a$.

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