Question

Explain Taylor series.

Answer 1

Taylor series:

The Taylor theorem expresses a function in the form of the sum of infinite terms. These terms are circumscribed from the derivative of a given function for a particular point. The standard definition of an algebraic function is presented using an algebraic equation. A function may be well illustrated by its Taylor series too. This series can also be used to determine various functions in lots of areas of mathematics

Taylor’s Series Theorem

Assume that if f(x) be a real or composite function, which is a differentiable function of a neighborhood number that is also real or composite. Then, the Taylor series describes the following power series :

$\begin{array}{l}f\left(x\right)=f\left(a\right)\frac{f^{\text{'}}\left(a\right)}{1!}(x-a)+\frac{f”\left(a\right)}{2!}(x-a{)}^2+\frac{f^{\left(3\right)}\left(a\right)}{3!}(x-a{)}^3+\dots .\end{array}$

In terms of sigma notation, the Taylor series can be written as

$\begin{array}{l}\sum _{n=0}^{\mathrm{\infty }}\frac{f^n\left(a\right)}{n!}(x-a{)}^n\end{array}$

Where f⁽ⁿ⁾ (a) = n^th derivative of f n!

= factorial of n.