The Taylor Series

Limits & The Taylor Series

Finding Coefficients with Taylor Polynomials

So here is a sample problem:

f(x)= x^7-x^2

Sources

Radius of Convergence

http://mathworld.wolfram.com/TaylorsTheorem.html

What is the coefficient for the term containing (x+4)^4 in the Taylor Polynomial, centered at x+=-4, of f?

Please follow along as I go over this problem on the board...

This is something that comes into play when you have a Taylor polynomial function and you want to see where another function lines up with it graphically at a certain set of value ranges. See graph for example

https://www.khanacademy.org/math/calculus-home/series-calc/taylor-series-calc/v/error-or-remainder-of-a-taylor-polynomial-approximation

http://www.dcs.warwick.ac.uk/people/academic/Steve.Russ/cs131/NOTE26.PDF

https://www.khanacademy.org/math/calculus-home/series-calc/taylor-series-calc/v/maclauren-and-taylor-series-intuition

http://www.sosmath.com/calculus/tayser/tayser01/tayser01.html

https://www.coursera.org/learn/advanced-calculus/lecture/OmGzx/how-do-taylor-series-provide-intuition-for-limits

Analytic Functions In Relation to the Taylor Series

What is the purpose of using the Taylor series to find a limit?

https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-2-applications-of-differentiation/part-c-mean-value-theorem-antiderivatives-and-differential-equations/session-34-introduction-to-the-mean-value-theorem/MIT18_01SCF10_ex34sol.pdf

Real analytic functions have the property that if you write down the Taylor Series, they will converge near that point to the function.

Taylor Series gives us more insight on a limit than just knowing what it actually is. For example, we know that the limit of the function above is 1/2. But the Taylor Series really lets us know why it's that way. The Taylor Series is really easy to understand that is we have a polynomial on the top, and a polynomial on the bottom, we just take the ratio of that to get our limit. "The point of math is Proof not Truth"

Suppose that f you can compute f^n(0). No matter what n may be.

We can say that the function is real analytic to it's Taylor series around zero. Real analytic means that the function HAS a power series representation at zero. We can also replace zero with a to generalize it.

What is the Taylor Series for f centered around a?

Suppose that f you can compute f^n(0). No matter what n may be. This can be represented by the following Taylor Series. But we don't know whether this series converges when x is around 0.

We're using the same power series as before but plugging in a for x instead of 0. We can now differentiate each term to find the next coefficient.

We begin to do the same thing where we plug in different terms and differentiate. This is the same pattern as the last one, so that means that we can come up with a new formula.

What is the Taylor series for F around zero

F is represented by a power series.

When you plug zero in for x, everything is canceled out except a sub zero. That means that a sub zero is has the same value of the function at zero. a sub 1 can be calculated by the derivative of f. We can differentiate the power series term by term

In order to find what the first term is, we take the derivative of the first problem. We can keep differentiating until we find all the terms we want to. As long as we have a power series representation for the function, we can find out all other terms.

This general rule can be made, but only if the function is shown by a power series. This can be called the Maclaurin series or the Taylor series centered at zero.

Linear approximation

In calculus 1 we were finding a linear approximation of f(x) around a. g(x) is the linear approximation (the right side of the equation). Here we have a function g who's value @ a is equal to value of f(a) and they have the same derivative.

f(x)= sin x

Using the same method as before, we're going to take the function sin x and differentiate it. Then we're going to find the 1st, 2nd, and 3rd term.

Here we have the same linear approximation added with a second term. The second term (the second derivative portion) makes the last part true, that g(a) will share the same second derivative as f(a)

With the same approximation formula we used before, we're able to get a cubic polynomial that approximates sin(x). Which it does approximate it well. This can be shown graphically and numerically

We know that polynomial approximation is better than linear approximation. And power series approximation is even better than polynomial approximation.

This picture shows us how the function g(x) has the same value of f(a) and the same 1st, 2nd, and 3rd derivatives. This is all figured out through differentiating.

The Taylor Series

Introduction

So far, we've been able to look at a power series and figure out a function that the power series represents. Now we want to flip that. With the Taylor Series, we want to take a function and figure out a power series that represents the function.

What is the Taylor Series around Zero

It's ridiculously easy to differentiate sin over and over again because there is a pattern that occurs.

How the Taylor Theorem & Mean Value Theorem relate to one another

The Mean Value Theorem comes in handy when trying to estimate the remainders/errors when you are playing around with Taylor Polynomials

The Taylor Theorem

The Mean Value Theorem gives you info on the first derivative

The Taylor Theorem tells you info on higher derivatives

What Makes A Taylor's Theorem a Taylor's Theorem?

Well first off, a Taylor Theorem has to meet certain requirements to be regarded as a Taylor Series.

The Taylor Theorem is also supposed to help with approximating functions using polynomials. The main functions that will benefit from this theorem will take the form of exponential functions, logarithms, and plain old sin.

The Taylor Theorem can be expressed using the following formula