Examples Using Taylor Series Formula
Example 1: Find the expansion for the function, f(x) = x3 centred at a = 2 using the Taylor Series Formula.
Solution:
We know that Taylor Series Formula is
f(x) = f(a) + [f'(a)’/1!](x-a) + [f”(a)/2!](x-a)2 + [f”'(a)/3!](x-a)3 +…+ f(n)(x)/n!(x-a)n
Given function, f(x) = x3
Centered at a = 2
f'(x) = 3x2
f”(x) = 6x
f”'(x) = 6
f””(x) = 0
Now the Taylor Series expansion of f(x) = x3 is,
f(x) = f(2) + [f'(2)’/1!](x-2) + [f”(2)/2!](x-2)2 + [f”'(2)/3!](x-2)3 + f””(a)/4!(x-2)4
f(x) = 8 + 12(x-2) + 6(x-2)2 + (x-2)3 + 0
f(x) = (x-2)3 + 6(x-2)2 + 12x – 16
Example 2: Find the expansion for the function, f(x) = 4x centred at a = 1 using the Taylor Series Formula.
Solution:
We know that Taylor Series Formula is
f(x) = f(a) + [f'(a)’/1!](x-a) + [f”(a)/2!](x-a)2 + [f”'(a)/3!](x-a)3 +…+ f(n)(x)/n!(x-a)n
Given function, f(x) = 4x
Centered at a = 1
f'(x) = 4
f”(x) = 0
Now the Taylor Series expansion of f(x) = 4x is,
f(x) = f(1) + [f'(1)’/1!](x-1) + [f”(1)/2!](x-1)2
f(x) = 4 + 4(x-1) + 0
f(x) = 4x
Example 3: Find the Taylor series expansion for function, f(x) = sin x, centred at x = 0.
Solution:
f(x) = sin x
Using Taylor series formula,
f(x) = f(a) + f′(a)(x − a) + f′′(a)/2! × (x − a)2+ f′′′(a)/3! × (x − a)3+ f(4)(a)/4! × (x − a)4+ … + f(n)(a)/n! × (x − a)n
Derivatives of the six x are,
f(x) = sin (x)
f'(x) = cos (x)
f”(x) = -sin (x)
f”'(x) = -cos (x)Now,
sin(x) = sin(a)+cos(a)/1! (x – a)−sin(a)/2! (x – a)2-cos(a)/3! (x – a)3+ …
Puttinga = 0
sin(x) = 0+1/1! (x – 0)−0/2! (x- 0)2– 1/3! (x – 0)3+ …
sin(x) = 1/x – 1/6x3 + …
Taylor Series
Taylor Series is the series which is used to find the value of a function. It is the series of polynomials or any function and it contains the sum of infinite terms. Each successive term in the Taylor series expansion has a larger exponent or a higher degree term than the preceding term. We take the sum of the initial four, and five terms to find the approximate value of the function but we can always take more terms to get the precise value of the function.
In this article, we will learn about the Taylor Series expansion, formula, Maclaurin Series, and others in this article.
Table of Content
- Taylor Series Expansion
- Taylor Series & Maclaurin Series
- Taylor’s Series Formula
- Taylor Series Theorem Proof
- Taylor Series of Sin x
- Taylor Series of Cos x
- Taylor Series in Several Variables
- Maclaurin Series
- Applications of Taylor Series