Error+Analysis


 * Error Analysis: Alternating Series Error, Integral Technique, and Lagrange Error Bound **

So on this wiki we're going to talk about finding the error of a series. First for clarification, the tail of a series is the sum of the terms in a series beyond a certain partial sum. The remainder is the value of the tail after a certain partial sum, given that the series converges. For alternating series, an upper bound for the entire tail of the series is determined by the first term (this makes sense if you think about it because the terms decrease in absolute value and thus the entire tail is less than the first term of the tail). For improper intergrals, you find the upper and lower bounds for the tail of a series with positive terms. And finally the Lagrange form of the remainder can be found to determine the number of terms needed to approximate the limit a series converges to to a specific accuracy. Hopefully that made some sense but bear with me, we'll look at each specifically and it'll just be super fun.

** Alternating S **** eries Error ** This is probably the easiest type of error analysis in that you basically just have to find the next term of the series (the first term of the tail). So the error of the partial sum of an alternating series to n terms, is. It's as simple as that! Here's an example:

Given the series:, using the partial sum S100 what is the maximum possible error of this approximation?

To solve this you would simply plug in n+1 (101) into n and you get 1/101 or 0.688

Another example would be:

Here's a video of a problem you might encounter (ignore the voice): media type="youtube" key="gYpBUQ_DnC8" height="385" width="640"

** Ok so for the integral technique you are basically integrating from the last term of the partial sum to infinity to give you an upper bound of the tail of the series which is the remainder. So in this example, the tail of the series after the partial sum of n=20 is shown. Integrating from 20 to infinity gives you an upper bound of the remainder and integrating from 21 to infinity gives you the lower bound.
 * Integral Technique

Lagrange Error Bound ** This one is a bit more complicated but still not that bad. First of all, we'll use the general term of the Taylor series expansions of f(x) about x=a which is: so then the first term of the tail would be:. This equation would give you the exact remainder but since it is so difficult to calculate, we use an upper bound approximation of the remainder using M, the maximum value of on [a,x] (a is where the equation is centered and x is where you're evaluating). Now your equation for the Lagrange error bound is:. Here's an example...

Consider the Mclaurin series So we know that e < 3 and therefore and our value of M is 9. Then we can just substitute into our equation for the error bound. Which means that:
 * 1. Estimate [[image:ww.jpg]] using the 11th partial sum (n=11)**
 * 2. Use the Lagrange form of the remainder to estimate the accuracy of using this partial sum.**

Here's another:



Sources: //Calculus: Concepts and Applications// by Paul Foerster [|Error Estimation] [|Lagrange Error] media type="custom" key="6402007"