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 well 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.

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.       

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 don’t have the actual value of the series.  However, we can use some of the tests that we’ve got for convergence to get a pretty good estimate of the remainder provided we make some assumptions about the series.  Once we’ve got an estimate on the value of the remainder we’ll also have an idea on just how good a job the partial sum does of estimating the actual value of the series.

There are several tests that will allow us to get estimates of the remainder.  We’ll go through each one separately.

Integral Test

Recall that in this case we will need to assume that the series terms are all positive and will eventually be decreasing.  We derived the integral test by using the fact that the series could be thought of as an estimation of the area under the curve of  where .  We can do something similar with the remainder.

First, let’s recall that the remainder is,

Now, if we start at , take rectangles of width 1 and use the left endpoint as the height of the rectangle we can estimate the area under  on the interval  as shown in the sketch below.

We can see that the remainder, R_n, is exactly this area estimation and it will over estimate the exact area.  So, we have the following inequality.

Next, we could also estimate the area by starting at , taking rectangles of width 1 again and then using the right endpoint as the height of the rectangle.  This will give an estimation of the area under  on the interval .  This is shown in the following sketch.

Again, we can see that the remainder, R_n, is again this estimation and in this case it will underestimate the area.  This leads to the following inequality,

Combining (2) and (3) gives,

So, provided we can do these integrals we can get both an upper and lower bound on the remainder.  This will in turn give us an upper bound and a lower bound on just how good the partial sum, s_n, is as an estimation of the actual value of the series.

In this case we can also use these results to get a better estimate for the actual value of the series as well.

First, we’ll start with the fact that

Now, if we use (2) we get,

Likewise if we use (3) we get,

Putting these two together gives us,

This gives an upper and a lower bound on the actual value of the series.  We could then use as an estimate of the actual value of the series the average of the upper and lower bound.

Let’s work an example with this.

So, that is how we can use the Integral Test to estimate the value of a series.  Let’s move on to the next test.

Comparison Test

In this case, unlike with the integral test, we may or may not be able to get an idea of how good a particular partial sum will be as an estimate of the exact value of the series.  Much of this will depend on how the comparison test is used.

First, let’s remind ourselves on how the comparison test actually works.  Given a series  let’s assume that we’ve used the comparison test to show that it’s convergent.  Therefore, we found a second series  that converged and  for all n.

What we want to do is determine how good of a job the partial sum,

will do in estimating the actual value of the series .  Again, we will use the remainder to do this.  Let’s actually write down the remainder for both series.

Now, since  we also know that

When using the comparison test it is often the case that the b_n are fairly nice terms and that we might actually be able to get an idea on the size of T_n.  For instance, if our second series is a p-series we can use the results from above to get an upper bound on T_n as follows,

Also, if the second series is a geometric series then we will be able to compute T_n exactly.

If we are unable to get an idea of the size of T_n then using the comparison test to help with estimates won’t do us much good.

Let’s take a look at an example.