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.
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) = nth derivative of f
n! = n factorial