Taylor Series
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About this Free Certificate Course
Taylor Series is the infinite sum of the terms expressed as the function's derivatives at any given point. This fundamental concept finds its usage in the domain of computers, calculus, chemistry, and even physics! It is one of the most structured ways to understand how we could estimate what a function would look like. This concept is equally important if you are working on mathematical concepts for your college work. This course has been designed to ensure that you have access to understanding the theory of how the Taylor series works and complementing this with ample examples to round your learning experience nicely.
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Frequently Asked Questions
What is a Taylor Series used for?
Taylor series can be applied to various concepts in the field of mathematics, computers, calculus, chemistry, and physics. A few of them are:
- Computing 7th degree Maclaurin polynomial
- Computing second-order Taylor series expansion of a function around the origin
- Finding Taylor series at some point of a function
- Indirect Taylor series expansion
How do you find the Taylor Series of a Function?
To find Taylor series of any function, apply these simple steps:
Step 1: Calculate the values of the first few derivatives of f(x).
Step 2: Evaluate the function and its derivatives at x=a.
Step 3: Fill the right-hand side of the Taylor series expression.
Step 4: Using a summation, write the final result.
Other methods include solving by its very definition, substitution, multiplication or division, addition or subtraction, applying integration by parts repeatedly, and computer algebra systems.
How do you find the sum of a Taylor Series?
The Taylor series is a sum of polynomials, which can be integrated term by term using the standard technique of integration. For the general form, integrate it as a normal polynomial because the only x is variable, n is a constant so all terms other than x are constants
∫∑∞n=0(−1)n+1(n+1).3n(x−2)ndx=∑∞n=0(−1)n+1(n+1).3n(x−2)n+1n+1+C
What is the condition for Taylor Series?
According to the Taylor theorem, any function f(x) that satisfies certain conditions can be expressed in the form of the Taylor Series.
fn(0) where (n = 1, 2, 3,……) is finite and |x| < 1.
The term f(n)(0)n! xn becomes less significant in contrast when n is small.
The contracts are determined by initial conditions and n (an integer) for the number of constraints is applied to the function.
Do Taylor Series always converge?
A simple answer to this is “NO”, but every Taylor series function converges at one point where x = a. The Taylor series has divergence and generalization as well.
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Taylor Series
Taylor Series of a function is an expansion of the function f(x) into an infinite sum of terms. These terms have larger exponents like x, x^2, x^3, etc. The Taylor Series formula can be denoted as sigma notation and its expansion. For example, sigma notation n=0xn/n! can be written in Taylor Series expansion as ex= 1+ x+ x2/2! + x3/3!+... and there are many more functions that can be denoted in the form of the Taylor Series. Taylor expansion is also an important concept that one must be aware of. You can get the approximate value for a function with the help of the first few terms of the Taylor formula. You can see the correct use of Taylor approximation in Euler’s formula for complex numbers.
Taylor and Maclaurin Series are pretty famous among many fields. When the ‘a’ in the Taylor Series is zero, it becomes a Maclaurin Series. Taylor formula is considered the key that allows one to provide an equation for the polynomial expansion for every function called f. Taylor Series helps you to approximate ugly functions into polynomials. Taylor Series is a mathematical concept utilized in all the other fields to get their solution to their problem. Taylor Series is also used in Data Science to deduce some algorithms of Machine Learning. Many other areas utilize it. Many of the most famous equations for Physics are derived with the help of the Taylor Series, Taylor polynomial formula, and Taylor Series expansion formula. Commo Taylor Series is taught to students in their colleges for solving Mathematical problems.
The formula k=0nfk(a)/k! (x-a)k = f(a) + f'(a)(x-a) + f''(a)/2! (x-a)2 + ... + f(n)(a)/n!approximates the function f(x) near ‘a’ and Taylor’s theorem provides the bounds for the error in this approximation. PN(x) = f(0) + f'(0)x + f''(0)/2! x2 + ... + f(n)(0)xn/n!is the Talor Polynomial of degree n=0. It is also called as the Maclaurin Series of degree n. When x=a for a function ‘f’ which has n+1 continuous derivatives it provides you with Taylor approximation formula.
Examples of Taylor Series:
For f(x) = e^x
fk(x) = ex
fk(0) = 1
Hence,
ex = 1 + x + x2/2! + x3/3! + ... + k=0xk/k!
For nRn(x) = necx(n+1)/(n+1)! = 0 for all c between 0 and x.
There are many real-world applications of the Taylor Series. The calculator you use can simple do addition, subtraction, multiplication, and division, but when you try to find sin x, the calculator starts computing. For this computing purpose, it makes use of the Maclaurin Series. Many of the Physics formulas are derived with the help of the Taylor Series, and it also plays a pivotal role in deriving many algorithms. Hence, Taylor Series is an important concept of Mathematics. If you wish to learn Taylor Series from scratch and understand it better, enroll in Great Learning’s Taylor Series free course and attain free certification on completing the course successfully.