A Gentle Introduction to Taylor Series

A Gentle Introduction to Taylor Series

A Gentle Introduction to Taylor Series

Taylor series expansion is an superstitious concept, not only the world of mathematics, but moreover in optimization theory, function propinquity and machine learning. It is widely unromantic in numerical computations when estimates of a function’s values at variegated points are required.

In this tutorial, you will discover Taylor series and how to injudicious the values of a function virtually variegated points using its Taylor series expansion.

After completing this tutorial, you will know:

  • Taylor series expansion of a function
  • How to injudicious functions using Taylor series expansion

Let’s get started.

A Gentle Introduction To Taylor Series. Photo by Muhammad Khubaib Sarfraz, some rights reserved.

A Gentle Introduction To Taylor Series. Photo by Muhammad Khubaib Sarfraz, some rights reserved.

Tutorial Overview

This tutorial is divided into 3 parts; they are:

  1. Power series and Taylor series
  2. Taylor polynomials
  3. Function propinquity using Taylor polynomials

What Is A Power Series?

The pursuit is a power series well-nigh the part-way x=a and unvarying coefficients c_0, c_1, etc.

What Is A Taylor Series?

It is an wondrous fact that functions which are infinitely differentiable can generate a power series tabbed the Taylor series. Suppose we have a function f(x) and f(x) has derivatives of all orders on a given interval, then the Taylor series generated by f(x) at x=a is given by:

The second line of the whilom expression gives the value of the kth coefficient.

If we set a=0, then we have an expansion tabbed the Maclaurin series expansion of f(x).

Examples Of Taylor Series Expansion

Taylor series generated by f(x) = 1/x can be found by first differentiating the function and finding a unstipulated expression for the kth derivative. 

The Taylor series well-nigh various points can now be found. For example:

Taylor Polynomial

A Taylor polynomial of order k, generated by f(x) at x=a is given by:

For the example of f(x)=1/x, the Taylor polynomial of order 2 is given by:

Approximation via Taylor Polynomials

We can injudicious the value of a function at a point x=a using Taylor polynomials. The higher the order of the polynomial, the increasingly the terms in the polynomial and the closer the propinquity is to the very value of the function at that point.

In the graph below, the function 1/x is plotted virtually the point x=1 (left) and x=3 (right). The line in untried is the very function f(x)= 1/x. The pink line represents the propinquity via an order 2 polynomial.

The very function (green) and its propinquity (pink)

The very function (green) and its propinquity (pink)

More Examples of Taylor Series

Let’s squint at the function g(x) = e^x.  Noting the fact that the kth order derivative of g(x) is moreover g(x), the expansion of g(x) well-nigh x=a, is given by:

Hence, virtually x=0, the series expansion of g(x) is given by (obtained by setting a=0):

The polynomial of order k generated for the function e^x virtually the point x=0 is given by:

The plots unelevated show polynomials of variegated orders that estimate the value of e^x virtually x=0. We can see that as we move yonder from zero, we need increasingly terms to injudicious e^x increasingly accurately. The untried line representing the very function is hiding overdue the undecorous line of the approximating polynomial of order 7.

Polynomials of varying degrees that injudicious e^x

Taylor Series In Machine Learning

A popular method in machine learning for finding the optimal points of a function is the Newton’s method. Newton’s method uses the second order polynomials to injudicious a function’s value at a point. Such methods that use second order derivatives are tabbed second order optimization algorithms. 

Extensions

This section lists some ideas for extending the tutorial that you may wish to explore.

  • Newton’s method
  • Second order optimization algorithms

If you explore any of these extensions, I’d love to know. Post your findings in the comments below.

Further Reading

This section provides increasingly resources on the topic if you are looking to go deeper.

Tutorials

Resources

Books

  • Pattern recognition and machine learning by Christopher M. Bishop.
  • Deep learning by Ian Goodfellow, Joshua Begio, Aaron Courville.
  • Thomas Calculus, 14th edition, 2017. (based on the original works of George B. Thomas, revised by Joel Hass, Christopher Heil, Maurice Weir)
  • Calculus, 3rd Edition, 2017. (Gilbert Strang)
  • Calculus, 8th edition, 2015. (James Stewart)

Summary

In this tutorial, you discovered what is Taylor series expansion of a function well-nigh a point. Specifically, you learned:

  • Power series and Taylor series
  • Taylor polynomials
  • How to injudicious functions virtually a value using Taylor polynomials

Do you have any questions?

Ask your questions in the comments unelevated and I will do my weightier to answer



Author: Shantun Parmar

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