Explain the Taylor series.

Solution:

The Taylor series is also represented in the form of functions of several variables. If the value of point ‘a’ is zero, then the Taylor series is also called the Maclaurin series.

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 in sigma notation can be written as

∑ = f⁽ⁿ⁾ (a) (x−a)ⁿ / n!

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

n! = n factorial

Thus, 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! +  ….

Explain Taylor series.

Summary:

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! +  ….