Taylor Series Formula

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The Taylor series formula is the representation of any function as an infinite sum of terms. These terms are calculated from the values of the function’s derivatives at a single point. This concept was formulated by the Scottish mathematician James Gregory. But, it was formally introduced by the English mathematician Brook Taylor in 1715. For the expansion of mathematical series, it is used extensively. In this topic, we will see the concept of Taylor series and Taylor Series Formula with examples.


What is Taylor Series?

A function can be approximated by using a finite number of terms in its Taylor series. Taylor’s theorem is providing quantitative estimates on the error. The polynomial formed by taking some initial terms of the Taylor series is popular as Taylor polynomial.

The Taylor series of a function is the limit of that function’s Taylor polynomials with the increase in degree if the limit exists. A function may not be equal to its Taylor series, although its Taylor series converges at every point.

We have two assumptions. First, assume that the function f(x) does, in fact, have a power series representation about x = a,

\[f(x) = \sum_{n=0}^{\infty} c_{n} (x - a)^{n} = c_{0} + c_{1}(x - a) + c_{2}(x - a)^{2} + c_{3}(x - a)^{3} +\] . . . .

Next, we will assume that the function, f(x), has derivatives of every order and that we can find easily.


The Formula of Taylor Series

\[f(x) = f(a) + f’(a)(x - a) + \frac{f''(a)}{2!} (x - a)^{2} + \frac{f'''(a)}{3!}(x - a)^{3} +\] . . . .


To find out a condition that must be true in order for a Taylor series to exist for a function, we first define the nth degree Taylor polynomial equation of


f(x) as, \[ T_{n}(x) = \sum_{i=0}^{n} \frac{f^{(i)} (a)}{i!} (x - a)^{i}\]


This polynomial is of degree at most n. If we have to write some without the summation notation this would clearly be an nth degree polynomial.


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Maclaurin Series Expansion

If the Taylor Series is centered at 0, then the series is known as the Maclaurin series. It means that, 

If a= 0 in the Taylor series, then we get,

\[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!} x^{2} + \frac{f'''(0)}{3!} x^{3} +\] ...... is known as Maclaurin Series Expansion.


Applications of Taylor Series

The uses of the Taylor series are:

  • Taylor series is used to evaluate the value of a whole function in each point if the functional values and derivatives are identified at a single point.

  • The representation of the Taylor series reduces many mathematical proofs.

  • The sum of partial series can be used as an approximation of the whole series.

  • Multivariate Taylor series can be used in many optimization techniques.

  • This series is used in the power flow analysis of electrical power systems.

Taylor Series Steps

Step 1: Calculate the first few derivatives of f(x).

We see in the taylor series general taylor formula, f(a). This is f(x) evaluated at x = a. Then, we see f '(a). This is the first derivative of f(x) evaluated at x = a.

Step 2: Evaluate the function and its derivatives at x = a.

Take each of the results from the previous step and substitute a for x.

Step 3: Fill in the right-hand side of the Taylor series expression, using the Taylor formula of Taylor series we have discussed above :

Using the Taylor formula of Taylor series:-

\[f(x) = f(a) + f’(a)(x - a) + \frac{f''(a)}{2!} (x - a)^{2} + \frac{f'''(a)}{3!}(x - a)^{3} +\] ....

Step 4: Write the result using a summation.

Having a summation of a general term will be useful when determining the interval of convergence, or the set of x-values where a series converges.


Where is the Taylor Series Used?

The Taylor Series is used in the power flow analysis of electrical power systems (Newton-Raphson method). Multivariate Taylor series can be used in different optimization techniques; this means that is you approximate your function as a series of linear or quadratic forms, and then successively iterate on them to find the optimal value.


Questions to be Solved

Question 1) Determine the Taylor series at x = 0 for f(x) = \[e^{x}\]

Solution) Given: f(x) = \[e^{x}\]

Differentiate the given equation,

f’(x) = \[e^{x}\]

f’’(x) =\[e^{x}\]

f’’’(x) = \[e^{x}\]

At x = 0, we get

f’(0) = \[e^{0}\] =1

f’’(0) = \[e^{0}\] =1

f’’’(0) = \[e^{0}\] = 1

When Taylor series at x = 0, then the Maclaurin series is

\[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!} x^{2} + \frac{f'''(0)}{3!} x^{3} +\]

\[e^{x} = 1 + x(1) + (\frac{x^{2}}{2!})(1) + (\frac{x^{3}}{3!})(1) +\].....

Therefore, \[e^{x} = 1 + x(1) + (\frac{x^{2}}{2!}) + (\frac{x^{3}}{3!}) +\].....

FAQ (Frequently Asked Questions)

Question 1) What is the Difference Between the Taylor Series and Maclaurin Series?

Answer) Maclaurin and Taylor Series. The Taylor series equation, or Taylor polynomial equation, is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. A Maclaurin Polynomial is a special case of the Taylor polynomial equation, that uses zero as our single point.

Question 2) Why do we Need Taylor Series?

Answer) Taylor Series are studied because polynomial functions are easy and if one could find a way to represent complicated functions as Taylor series equation (infinite polynomials) then one can easily study the properties of difficult functions.

Question 3) Does Taylor Series Always Converge?

Answer) Because the Taylor series is a form of power series, every Taylor series equation also has an interval of convergence. ... However, when the interval of convergence for a Taylor series is bounded - that is when it diverges for some values of x - you can use it to find the value of f(x) only on its interval of convergence.

Question 4) Who Invented the Taylor Series?

Answer) Brook Taylor

As far as I know, the concept of Taylor series was discovered by the Scottish mathematician James Gregory and formally introduced by the English mathematician Brook Taylor in 1715. However, my main curiosity is about the problems and situations that resulted in a need to approximate a function using the Taylor series.