Taylor Series Formula: Meaning, Theorem, Steps, Application

Taylor Series Formula can be defined as the illustration of any function as an infinite sum of terms. These terms are computed from the values of the function’s derivatives at a single point. The idea of the Taylor series was formulated by James Gregory who was a Scottish mathematician. But, it was formally introduced in the year 1715 by an English mathematician Brook Taylor. It is extensively used for the elaboration of mathematical series. A function can be approximated by using a finite number of terms of its Taylor series. Taylor’s theorem provides quantitative estimations on the error which were introduced by the usage of such an approximation. 

Key Terms: Taylor Series Formula, Derivatives, Mathematical Series, Approximation, Quantitative Estimation, Polynomials, Convergence, Matrix, Notation, Logarithms, Operators, Constraints, Multivariate Taylor Series, Maclaurin Series Expansion

Also read: Isosceles Triangle Theorems

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What is Taylor Series?

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In Mathematics, the Taylor series of a function is an indefinite total of terms represented in forms of the function's derivatives at a single point. Primarily, for typical functions, the function and the sum of its Taylor series are equivalent about this point. 

A Taylor series is also named a Maclaurin series if at any point the derivatives are considered 0. In other terms, the Taylor series of a function is the limitation of that function’s Taylor polynomials as the degree increases, provided that the limit exists. Even if at every point, the Taylor series of a function converges, the function might not be equivalent to its Taylor series. A function that is equivalent to its Taylor series in an open interval is termed as an analytic function in that interval.

We can take two hypotheses. First, suppose that the function f(x) does have a power series expressed about x=a,

Next, we will consider that the function, f(x), has derivatives of every order and that we can discover efficiently.

Taylor Series

Also Read: Applications of Derivatives

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Formula for Taylor Series

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Taylor Series is used to illustrate functions and "operators" in various regions of mathematics. In particular, this is valid in regions where the classical descriptions of functions break down. For instance, using the Taylor series, one may expand analytic functions to sets of matrices & operators, such as the matrix exponential/matrix logarithm.

To discover a condition that must be true for a Taylor series to exist for a function, we foremost specify the nth degree Taylor polynomial equation of

The above polynomial is of degree at most n. If we have to write some without the summation notation then it would be an nth degree polynomial.

Also Read: Differential Equation

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Theorem For Taylor Series

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According to Taylor's theorem, any function, say f(x) which satisfies specific conditions can be represented as a Taylor series. Taylor's theorem provides quantitative estimations on the error introduced by the use of approximations. 

We now have the following:

On |x-a| < R

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Maclaurin Series Expansion

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When the Taylor Series is centred at 0, then the series is referred to as the Maclaurin series. This implies that if we assume the value of a= 0 in the Taylor series, then we will obtain the following outcome,

The above equation is termed as Maclaurin Series Expansion.

In other words, a Maclaurin series is a power series that helps to compute an approximation of a function f(x) for input values close to 0, given that one knows the values of the successive derivatives of the function at 0. In multiple practical applications, it is equivalent to the function it defines.

Taylor and Maclaurin Series

Also Read: List of Integral Formulas

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Steps Involved in the Taylor Series

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The steps involved in determining the Taylor series have been mentioned below:

• Step 1: In the first step, compute the first few derivatives of f(x).

In the Taylor series general Taylor formula, f(a). This is f(x) which can be evaluated at x = a. Then, we see f '(a) which is the first derivative of f(x) evaluated at x = a.

• Step 2: Now, estimate the function and its derivatives at x = a.

Consider each outcome from the earlier step and replace a for x.

• Step 3: Using the Taylor formula of the Taylor series we have discussed above, fill in the right-hand side of the Taylor series expression:

Using the Taylor formula of Taylor series:-

• Step 4: At last, write the outcome of the equation using a summation.

Including a summation of a general term will be helpful when specifying the interval of the intersection or the set of x-values where a series converges/intersects.

Also Read:

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Applications of Taylor Series

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The applications of the Taylor series have been mentioned in the list below:

• Taylor series is used for estimating the value of a complete function in each point if the functional values and derivatives are identified at a single point.
• The expression of the Taylor series reduces numerous mathematical explanations.
• The aggregate of partial series can be utilized as an approximation of the entire series.
• The multivariate Taylor series can be used in multiple optimization techniques.
• The Taylor series is also used in the power flow analysis of electrical power techniques.

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Things to Remember

• A Taylor Series is the expansion of a function into an indefinite sum of terms, where every term possesses a larger exponent such as x, x2, x3, etc. 
• It's a special type of power series specified only for the functions which are infinitely differentiable on some open interval. 
• Taylor's theorem provides an approximation of a k-times differentiable function about a given point by a polynomial of degree k, named the kth-order Taylor polynomial. For a smooth function, Taylor polynomial is described as the truncation at the order k of the Taylor series of the function.
• The elaboration or expansion of the Taylor series of a function around 0 is referred to asthe Maclaurin series
• All the functions which are infinitely differentiable at a given point 0, have a Taylor series at that point. In case a function is holonomic (analytic), then it follows the Cauchy Riemann equations.