Truncation Error Example: Summation of a Series

In this segment we'll talk about, uh, truncation error and we'll go through an example. So, the example which we have here is that we are calculating e^1.2 by using the infinite Maclaurin series to be able to calculate it. The first part is asking you to use only the first three terms of the series and the second part is asking you to use the first four terms of series to calculate the truncation error. So, let's look at part a. Part a tells us we're taking the first two, first three terms of the Maclaurin series the first three terms of the Maclaurin series will be as follows: so, in that case what we'll get is this will be the approximation and that approximation will turn out to be 2.92. So that is the approximate value of e^1.2. So, in this case the truncation error will be equal to the exact value or also called true value minus the approximate value which we just obtained. The exact value is obtained from the calculator, which we get as 3.3201, that’s what we get from our calculator for e^1.2, and the approximate value turns out to be 2.92. And this number here turns out to be 0.4001. So, we're assuming that whatever we're getting from the calculator is exact after the first five significant digits which I have written there, so we subtract the approximate value from it, and we get the truncation error as follows. Now for part b we are now going to use the first four terms of the Taylor series or Maclaurin series. And this turns out to be as follows: so, this particular four terms of the series gives us a value of 3.208. So, if we want to calculate our truncation error in this case it will be of course again the exact value minus the approximate value. The exact value has been obtained by using the calculator for e^1.2, which turns out to be 3.3201. The approximate value is 3.208, and this number here turns out to be 0.1121. So, there's a specific reason why I chose a problem to show you which has two parts to it: because I wanted to show you that hey as you take more terms in the series that you get better approximations; so, when we have 3 terms, e^1.2 turned out to be 2.92, with a corresponding truncation error of 0.4001. And when we had four terms in the series, e^1.2 turned out to be approximately equal to 3.208, and the truncation error was 0.1121. So, as you're finding out that as you take more and more terms in the series that you are going to get in a better and better approximation, of e^1.2 in this case. So, that's what we mean by truncation error that if you have a math procedure in this case math procedure is simply adding infinite number of terms of the series together. But we cannot afford to take infinite number of terms, so we're taking a finite number of terms. So, whatever is the difference between the two because you have now approximated a mathematical procedure by taking only finite number of terms, you get truncation error. And that's the end of this segment