Taylor series

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\frac{d}{dx} \sin x=?\, This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

The Taylor series of a function is useful for approximating a mathematical function near to some particular point. For a function f(x), the Taylor series about the point x0 is


where each of the derivatives is to be evaluated at x = x0. If as N\rightarrow\infty the series converges, then it is exact. Otherwise, it can be used as an approximation. Often, Taylor series are performed around x0 = 0, in which case they are sometimes also known as a Maclaurin series.

Examples of common Taylor series

e^x=\sum_{n=0}^{\infin} \frac{x^{n}}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots

\sin x=\sum_{n=0}^{\infin} (-1)^n\frac{x^{2n+1}}{(2n+1)!}=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots

\cos x=\sum_{n=0}^{\infin} (-1)^n\frac{x^{2n}}{(2n)!}=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots

Extensions of the Exponential Function

Consider the exponential of imaginary number yi,


= cosy + isiny

By the power laws then all complex numbers have an exponential,

ex + yi = ex(cosy + isiny).

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