Explanation

A taylor series is an infinite sum of polynomials used to approximate a function, but only near the point $a$ , which is the point the series is calculated around.

A taylor series works by calculating the derivatives of the original function, substituting $a$ for $x$ , then integrating these derivatives to create a polynomial that will eventually converge to the original function.

Not all taylor series will converge back to the original function, and some will only converge close to the point $a$ .

The equation for a taylor series is given by the following formula:

$$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

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Visualization