Estimating the Value of a Series

Estimating the Value of a Series - Estimating the Value of...

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Estimating the Value of a Series We have now spent quite a few sections determining the convergence of a series, however, with the exception of geometric and telescoping series, we have not talked about finding the value of a series. This is usually a very difficult thing to do and we still aren’t going to talk about how to find the value of a series. What we will do is talk about how to estimate the value of a series. Often that is all that you need to know. Before we get into how to estimate the value of a series let’s remind ourselves how series convergence works. It doesn’t make any sense to talk about the value of a series that doesn’t converge and so we will be assuming that the series we’re working with converges. Also, as we'll see the main method of estimating the value of series will come out of this discussion. So, let’s start with the series (the starting point is not important, but we need a starting point to do the work) and let’s suppose that the series converges to s . Recall that this means that if we get the partial sums, then they will form a convergent sequence and its limit is s . In other words, Now, just what does this mean for us? Well, since this limit converges it means that we can make the partial sums, s n , as close to s as we want simply by taking n large enough. In other words, if we take n large enough then we can say that, This is one method of estimating the value of a series. We can just take a partial sum and use that as an estimation of the value of the series. There are now two questions that we should ask about this. First, how good is the estimation? If we don’t have an idea of how good the estimation is then it really doesn’t do all that much for us as an estimation. Secondly, is there any way to make the estimate better? Sometimes we can use this as a starting point and make the estimation better. We won’t always be able to do this, but if we can that will be nice.
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So, let’s start with a general discussion about the determining how good the estimation is. Let’s first start with the full series and strip out the first n terms. (1) Note that we converted over to an index of i in order to make the notation consistent with prior notation. Recall that we can use any letter for the index and it won’t change the value. Now, notice that the first series (the n terms that we’ve stripped out) is nothing more than the partial sum s n . The second series on the right (the one starting at ) is called the remainder and denoted by R n . Finally let’s acknowledge that we also know the value of the series since we are assuming it’s convergent. Taking this notation into account we can rewrite (1) as, We can solve this for the remainder to get, So, the remainder tells us the difference, or error, between the exact value of the series and the value of the partial sum that we are using as the estimation of the value of the series. Of course we can’t get our hands on the actual value of the remainder because we
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This note was uploaded on 11/10/2011 for the course MATH 136 taught by Professor Prellis during the Fall '08 term at Rutgers.

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Estimating the Value of a Series - Estimating the Value of...

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