# Explain Taylor series

The Taylor series is also represented in the form of functions of several variables.

## Answer: The general form of Taylor series is given by, f(x) = f(a) + f'(a) (x−a) / 1! + f''(a) (x−a)^{2} / 2! + f'''(a) (x−a) / 3! + ….

If the value of point ‘a’ is zero, then the Taylor series is also called the Maclaurin series.

**Explanation:**

Taylor series is a series of an infinite sum of terms, which are often represented as a polynomial function. Each successive term has a larger exponent than the preceding one.

f (x) = f (a) f′ (a)1!(x−a) + f” (a) 2!(x−a)2 + f (3) (a)3!(x−a)3+….

The above Taylor series expansion is given for a real values function f(x) where f’ (a), f’’ (a), f’’’ (a), etc., denotes the derivative of the function at point a.

__Taylor 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 :

f(x) = f(a) + f'(a) (x−a) / 1! + f''(a) (x−a)^{2} / 2! + f'''(a) (x−a) / 3! + .....

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

∑ = f^{(n)}_{ }(a) (x−a)^{n} / n!

Where, f^{(n)} (a) = n^{th} derivative of f

n! = n factorial