Taylor Series by Enrique Mateus Nieves

Taylor Series by Enrique Mateus Nieves

Taylor series are mathematical representations of functions as infinite sums of terms derived from their derivatives at a specific point. Authored by Enrique Mateus Nieves, this work explores the history, applications, and convergence of Taylor series, including Maclaurin series. It is essential for students and professionals in mathematics and engineering seeking to understand function approximation and analytic functions. The document provides examples, historical context, and practical applications, making it a valuable resource for anyone studying calculus or mathematical analysis.

Key Points

  • Explains the concept of Taylor series and their applications in mathematics.
  • Covers the historical development of Taylor series, including contributions from mathematicians like Brook Taylor and Colin Maclaurin.
  • Includes practical examples of Taylor series for functions such as exponential, logarithmic, and trigonometric functions.
  • Discusses the convergence and properties of analytic functions related to Taylor series.
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Prof. Enrique Mateus Nieves
PhD in Mathematics Education.
Taylor series
As the degree of the Taylor polynomial rises, it approaches the correct function. This
image shows and Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11
and 13.
The exponential function (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red).
Prof. Enrique Mateus Nieves
PhD in Mathematics Education.
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that
are calculated from the values of the function's derivatives at a single point.
The concept of a Taylor series was formally introduced by the English mathematician Brook
Taylor in 1715. If the Taylor series is centered at zero, then that series is also called a
Maclaurin series, named after the Scottish mathematician Colin Maclaurin, who made
extensive use of this special case of Taylor series in the 18th century.
It is common practice to approximate a function by using a finite number of terms of its Taylor
series. Taylor's theorem gives quantitative estimates on the error in this approximation. Any
finite number of initial terms of the Taylor series of a function is called a Taylor polynomial. The
Taylor series of a function is the limit of that function's Taylor polynomials, provided that the limit
exists. A function may not be equal to its Taylor series, even if its Taylor series converges at
every point. A function that is equal to its Taylor series in an open interval (or a disc in the
complex plane) is known as an analytic function.
Definition: The Taylor series of a real or complex-valued function Æ’(x) that is infinitely
differentiable in a neighborhood of a real or complex number a is the power series
which can be written in the more compact sigma notation as
where n! denotes the factorial of n and Æ’
(n)
(a) denotes the nth derivative of Æ’ evaluated at the
point a. The derivative of order zero ƒ is defined to be ƒ itself and (x − a)
0
and 0! are both
defined to be 1. In the case that a = 0, the series is also called a Maclaurin series.
Examples
The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for (1 −
x)
−1
for |x| < 1 is the geometric series
so the Taylor series for x
−1
at a = 1 is
Prof. Enrique Mateus Nieves
PhD in Mathematics Education.
By integrating the above Maclaurin series we find the Maclaurin series for log(1 − x), where log
denotes the natural logarithm:
and the corresponding Taylor series for log(x) at a = 1 is
and more generally, the corresponding Taylor series for log(x) at some is:
The Taylor series for the exponential function e
x
at a = 0 is
The above expansion holds because the derivative of e
x
with respect to x is also e
x
and e
0
equals 1. This leaves the terms (x − 0)
n
in the numerator and n! in the denominator for each
term in the infinite sum.
History: The Greek philosopher Zeno considered the problem of summing an infinite series
to achieve a finite result, but rejected it as an impossibility: the result was Zeno's paradox. Later,
Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was
apparently unresolved until taken up by Democritus and then Archimedes. It was through
Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be
performed to achieve a finite result.
[1]
Liu Hui independently employed a similar method a few
centuries later.
[2]
In the 14th century, the earliest examples of the use of Taylor series and closely related
methods were given by Madhava of Sangamagrama.
[3][4]
Though no record of his work survives,
writings of later Indian mathematicians suggest that he found a number of special cases of the
Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and
arctangent. The Kerala school of astronomy and mathematics further expanded his works with
various series expansions and rational approximations until the 16th century. In the 17th
century, James Gregory also worked in this area and published several Maclaurin series. It was
not until 1715 however that a general method for constructing these series for all functions for
which they exist was finally provided by Brook Taylor,
[5]
after whom the series are now named.
The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published
the special case of the Taylor result in the 18th century.
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FAQs of Taylor Series by Enrique Mateus Nieves

What is the significance of Taylor series in mathematics?
Taylor series are crucial in mathematics as they allow functions to be approximated by polynomials, making complex calculations simpler. They are particularly useful in calculus for analyzing function behavior near a specific point. The series provide insights into the function's derivatives, enabling predictions about its values and behavior. Understanding Taylor series is essential for fields such as physics and engineering, where approximations are often necessary for practical applications.
How does the Taylor series relate to the concept of analytic functions?
An analytic function is one that can be represented by a Taylor series in a neighborhood around a point. This means that the function is equal to its Taylor series within that interval, allowing for accurate approximations. The document highlights that not all functions are analytic, as some may have Taylor series that converge but do not equal the function itself. This distinction is important in advanced mathematics, particularly in complex analysis.
What are some common examples of functions represented by Taylor series?
Common examples of functions represented by Taylor series include the exponential function, natural logarithm, and trigonometric functions like sine and cosine. For instance, the Taylor series for the exponential function e^x converges for all x and is used extensively in various applications. The document provides detailed expansions for these functions, illustrating how Taylor series can simplify calculations and enhance understanding of their properties.
What historical figures contributed to the development of Taylor series?
The development of Taylor series can be traced back to several key historical figures, including Brook Taylor, who formalized the concept in the early 18th century. Colin Maclaurin also made significant contributions by exploring special cases of Taylor series, which are now known as Maclaurin series. Earlier mathematicians, such as Madhava of Sangamagrama, laid foundational work in infinite series, influencing later developments in calculus.
How do Taylor series aid in function approximation?
Taylor series provide a powerful tool for approximating complex functions using polynomials, which are easier to compute. By taking a finite number of terms from the Taylor series, one can achieve a close approximation of the function within a specified range. This method is particularly useful in numerical analysis and computational mathematics, where exact solutions may be difficult to obtain. The document emphasizes the practical applications of Taylor series in various scientific and engineering fields.

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