- is a convergent geometric series.
- is a divergent geometric series.
- diverges by the divergence test.
For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.
We thus have two important sequences associated with any convergent series .
- : The term is found by adding all terms up to and including and tells us the approximate value of the infinite series .
- : The term is the error made by approximating the infinite series by (that is, by approximating by ).
When we have an explicit formula for , we can determine an exact formula for the error term ; we simply use the relationship . We can compute by taking and subtract form both sides to obtain the formula for . We have seen that there are two special types of series when we can find such a description for , which we explore below.
Recall that a series is geometric if for each , there are constants and for which . We also know precisely when such series converge.
En route to establishing this result, we determined that when ,
We can use this to find a formula for when .
We can also compute that (either directly from the above or from the convergence result for geometric series). We can now find a formula for .
Armed with an explicit formula for both and , we can arrange the first several terms in each sequence in a table.
Note that for each value of , we have that , which should not be surprising! We thus see that for each , gives the approximate value of the series, while gives the error of this approximation. Furthermore, notice something nice here; is a increasingdecreasing sequence, meaning that as we use more terms to approximate the infinite series, the error becomes smaller!
In most applications, we will only want to determine the value of a convergent series to a specified degree of accuracy. While we can compute the exact value of the series in the last example, this is not always the case. To gain some insight, we will use the previous exercise to estimate the series to within of its exact value. To do this, we will add finitely many terms and verify how good this approximation is.
Using a calculator (or other technology), we compute to one decimal place that . This means that we need to sum up to at least , meaning that we will need to use .
We thus know that will approximate to within at least of its exact value. To verify this, compute to four decimal places.
The other type of series for which we could find an explicit formula for are telescoping series.
We can write out several terms in the sequence of partial sums.
With both the value of and a formula for , we may find a formula for .
By using the formula for , what error is made by approximating by ?
When we have a convergent geometric or telescoping series, we can find an explicit formula for , which allows us to find the exact value of the series as well as an explicit formula for the terms in the sequence of remainders. We have then seen how the terms in the sequence of remainders give the error made by approximating the infinite series by a finite sum.
One very important point to remember is that the calculations in this section are possible because we consider convergent series; remember it is not possible to define a sequence of remainders for a divergent series!