Taylor+Series

**Taylor Series **  **A Taylor series expansion about a point x=a, is a power series expansion that’s useful to approximate the function in the neighborhood of the point x=a. **  **Taylor **** series are series using the sum of the terms whose components are: ** **(The nth derivative of f(x) at the center point, a, divided by n!) times (x-a)^n ** **In summation notation: **

   <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">However, every function does not have an appropriately fitting Taylor series function. The function must be differentiable for all values of its domain. Therefore, functions with a corner, cusp, vertical tangent, or a discontinuity, which are not differentiable at every point, will fail to have a Taylor series function. The most common Taylor series are those for functions whose derivatives will continue ad infinitum like (e^x and the trigonometric functions). ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">For example, the Taylor series expansion of sin x centered at x=0 will become more and more accurate as the order of the partial sum increases, until it finally matches the graph exactly as shown here. ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> ﻿ <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">[|Taylor Series Tutorial] **

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Remember that Linearization (or Euler’s Method) was L(x) = f(x) + f’(x)dx, which is basically the tangent line centered at some point a. Taylor series are very similar except as the order of the derivatives increases, the approximation is more accurate. In essence, linearization shows the first two terms of the Taylor series for that function centered at that point a. ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Just like when we used approximations for integrals using rectangles, we knew there must be a better way than using these shapes that gave us too much error. From LRAM, RRAM, and MRAM we moved into better approximations like Trapezoidal Method and Simpson's Method, which began to get closer to the curve by using tangent line and arcs. Just like this, we progress from the tangent line (linearization) to better approximations like Taylor Series. The taylor series uses higher order derivatives in order to better adjust to the behavior of the original function and come closer to the actual value of the function. We can now use quadratic and cubic or 10th and 11th order approximation lines that will be very near to the original function. ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">It is important to realize that Taylor series are still approximations. When calculating the estimated value of a function at some value, we want to compare that estimation with the actual value of the function at that point. One way of determining this error is by using the Remainder Estimation Theorem. Remainder Estimation uses the last term of the taylor series, the remainder, to show truncation error. This error estimation is also known as the Lagrange form. ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">[|Lagrange Error Applet] **

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**__What is the Talor Series Used For?__**

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**Now that you know how to find a Taylor Series, and when it can and can't be used, let's talk about what it can be used for.** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**Taylor Series can be used to find definite integral that otherwise wouldn't be evaluatable, like the following.** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">** ﻿ ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**This integral is not directly evaluatable, nor would u substitution be any help. Taylor Series comes in handy here. The Taylor Series for sin x is** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">
 * 1) Definite Itegrals **
 * Next, sub in for x**
 * If we antidifferentiate the result, and evaluate from 0 to 1, we get an answer that is a reasonable approximation of the original integral that we were trying to evaluate.**
 * 2) Growth of Functions**
 * Let's look at this example:**


 * for any exponent n. The ratio is measuring how large the exponential is compared to the polynomial. If this ratio was very small, we would conclude that the polynomial is larger than the exponential. But if the ratio is large, we would conclude that the exponential is much larger than the polynomial. The fact that this ratio becomes arbitrarily large means that the exponential becomes larger than the polynomial by a factor which is as large as we would like. This is what we mean when we say "an exponential grows faster than a polynomial." **
 * To see why this relationship holds, we can write down the Taylor series for [[image:taylor_6.JPG]] **

<span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">**__<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Example Problems __** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">
 * Notice that this last term becomes arbitrarily large as x approaches infinity. That implies that the ratio we are interested in does as well: **
 * Basically, the exponential grows faster than any polynomial because it behaves like an infinite polynomial whose coefficients are all positive. ** <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"> [[image:taylor_10.JPG width="576" height="810"]]

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