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Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples

Taylor Series, Coefficients & Polynomials: Definition, Equations & Examples
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  • 0:04 The Taylor Series
  • 0:29 Approximating a Polynomial
  • 2:42 Approximating…
  • 5:43 Lesson Summary
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Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

A Taylor series is an infinite series representing a function. In this lesson we explore how to use a finite number of terms of a Taylor series to approximate an integration result.

The Taylor Series

Sometimes a bread recipe is overly complicated. Maybe the bread will taste fine if we approximate some of the ingredients; like using ordinary butter instead of refined purified butter. In math, some complicated functions can be approximated. This is where the Taylor series is very effective. In this lesson, we'll learn about the Taylor series and that it's used to get an approximate answer to an otherwise impossible definite integration.

Approximating a Polynomial

A nice compact way to write the Taylor series uses the summation notation appearing below:


equation 1


Just for fun, let's expand this infinite sum out to n = 2 in which, as you can see, we get:


null


What if the f(x) on the left-hand side is the polynomial:


f(x)=4x^3+x-5


Then, f(a) is found by replacing x with a, as you can see here:


f(a)=4a^3+a-5


Next, we find the derivative, f ':


f_prime_(x)=12x^2+1


And replacing x with a again:


f_prime(a)=12a^2+1


Then we differentiate once more, with:


f_double_prime_(x)=24x


Substituting with a so it becomes:


f_double_prime_(a)=24a


The terms keep getting smaller. The next derivative is just 24 and the derivatives after this are all zero.

Now for the third derivative:


f_triple_prime(x)=24


Evaluating at x = a, as you can see, we get:


f_triple_prime(a)=24


Now for the fun. We have all the ingredients for the right-hand side, as you can see with all the values in these calculations:


the_right_hand_side


The right-hand side is exactly equal to the left-hand side. This isn't an approximation. But please resist the impulse to expand the right-hand side, which, let's be real, is what we're programmed to do. Instead, keep the terms organized in brackets. Also, let's do an approximation by keeping only the first three terms (the ones in blue). The polynomial we get by keeping some but not all of the terms is called a Taylor polynomial.

The left-hand side is a third-degree Taylor polynomial in x, as you can see in this graph:


There are two turns in f(x)
There_are_two_turns_in_f(x)


Our approximation is a second degree polynomial in x. This function turns only once. It's a parabola:


Approximating at a=2
Approximating_at_a=2


See how the approximation (the green curve) at x = 2 is really close to the polynomial (the blue curve)? We have a very good match at x = 2 because we evaluated the Taylor series at a = 2.

What if we wanted our approximation at x = -1?


The parabola (green curve) flips automatically
The_parabola_(green_curve)_flips_automatically


The Taylor series is really powerful. We can move the approximation to our point of interest and the series will automatically accommodate the best match.

Approximating Unsolvable Integrals

The Gaussian function appears in all types of applications like probability and image processing.


The bell-shaped Gaussian function
The_bell-shaped_Gaussian_function


If we integrate over all values of x we get a constant. This integral is surprisingly easy to do. What's even more surprising is there is no closed-form solution to this integral as an anti-derivative plus a constant. You can try all of your favorite methods like substitution, trig substitutions, and integration-by-parts. Nothing works. The best we have are approximations, usually using the error function. Time to apply the Taylor series to a real math application.

First, we write the integral of the Gaussian function in all its detail, as you can see below:


1/sqrt(2pi_sigma&2)_int_e^-(x-u)^2/2sigma^2_dx


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