Maclaurin Series
We know that the Taylor series 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
If the Taylor series is centred at x = 0, i.e. the value of f(x) is found at x = 0 then this series is called the Maclaurin Series.
Then the Maclaurin Series is,
f(x) = f(0) + [f'(0)’/1!](x) + [f”(0)/2!](x)2 + [f”'(0)/3!](x)3 +…+ f(n)(0)/n!(x)n
This above series is known as the Maclaurin series.
Example: Maclaurin series of ex is,
Solution:
We know that the Maclaurin series expansion is,
f(x) = f(0) + [f'(0)’/1!](x) + [f”(0)/2!](x)2 + [f”'(0)/3!](x)3 +…+ f(n)(0)/n!(x)n
The Maclaurin series of ex is,
ex = 1 + x + x2/2! + x3/3! + x4/4! + . . .
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