We can approximate the value of convergent series using the sequence of partial sums.

Given a series , our attempt to determine whether it converges or diverges requires that we consider the limit of the corresponding sequence of partial sums whose terms are given by . For geometric and telescoping series, we were able to find an explicit formula for the terms in the sequence of partial sums then analyze the limit directly. We have now seen that this is not always possible, but thankfully, we have developed a few tests that allow us to determine whether this limit exists. However, this leads to an important question.

Once we have established that a series converges, how can we determine its value?

If we aren’t able to find an explicit formula for , we likely will not be able to find the exact value of the series. For most practical applications that arise in science and engineering, we really do not need to find the exact value of a convergent series; we only need to know the value to within a predetermined margin of error.

Remainders

For a series , we may write the following.

Note that is precisely . Note that for every value of , must be finite since the sum that defines it is the sum of finitely many terms. Keeping this in mind, notice the following.

  • If diverges, then must divergecould converge or divergemust converge since the lower index of summation affectsdoes not affect whether a series converges or diverges .
  • If converges, then must divergecould converge or divergemust converge since the lower index of summation affectsdoes not affect whether a series converges or diverges .

Now, if the series converges, by definition. This means that the finite sum should become as close as we want to the value of the infinite series if we choose a large enough value of . This means that we can approximate the value of by using terms in the sequence of partial sums!

Now that we have an idea of how to approximate the value of a convergent series, the next important question to ask is: just how good is this estimate? This is where the idea of a remainder comes in.

Note that is also an infinite series, and its index starts at . The reason for this choice is that it allows us to split up a convergent series in a useful way.

The remainder as “error”

When we want to approximate the value of a convergent series by the finite sum , we generally want to be very “close” to the value of the infinite series where, “close” will vary from application to application. However, this idea gives a good conceptual interpretation of the remainder.

We thus can think about as the error made when approximating the value of the infinite series by the finite sum .

Remainders and convergence

In the last example, , and we can compute readily that . This leads to an interesting question.

If converges, must ?

The answer is yes! If converges, call its value . Since converges, by definition, we have that .

By considering the relationship and substituting , we may write

Taking limits of both sides gives us the desired result.

Thus, if converges, then we must have that . Conceptually, this should not be surprising since this means that as we use larger and larger values for , the terms approximate eventually approximate the value of the series more closely since the error will approach .

As it turns out, the converse of the statement in the last paragraph is also true, but showing it requires a more technical argument that relies on the formal definition of the limit of a sequence. Instead of providing the details, we state the following important theorem.

Summary

For a convergent series , we often cannot determine its exact value, but we can always write

and note the following.

  • For a given value of , is a finite sum and can be computed by hand (or by using technology). For a fixed value , is an approximation to the series . To obtain better approximations for in practice, we add more terms; since , this means that eventually, using higher values for will provide better approximations.
  • For a given value of , is an infinite sum and gives the error made when approximating by . Since , by using higher and higher values for , we can eventually make the error as close as we want to .

We finish the section with a few questions to check your understanding of what has been presented so far.

For a given infinite series, we can define a sequence of remainders
all the time, even when the series diverges. when the series converges. only when we have an explicit formula for the terms in the sequence of partial sums.
Select all of the following statements that are true.
If converges, then . It is possible to find a divergent series for which . It is possible to find a convergent series for which . Once we have established that a series converges, we can approximate its value by calculating the finite sum . Once we have established that a series converges, we can think of as the error made when approximating the infinite series by .