If an infinite sum converges, then its terms must tend to zero.
- Consider the associated sequence of partial sums, where .
- Try to find an explicit formula for the term . If you can find such a formula,
- If the limit exists, converges, and if we can determine that , then .
- If does not exist, then diverges.
- If an explicit formula for cannot be found, further analysis is needed.
In the previous section, we studied two types of series where we could find an explicit formula for , but unfortunately, this is not always easy or possible. Fortunately, it is not always necessary to do this in order to determine whether exists. Consider the example below.
We can observe also that , so eventually, the terms in this list become as close to as we want. So, eventually, we are trying to add together infinitely many numbers together that are very close to . Such an attempt cannot produce a finite result, so we expect the series to diverge.
Of course, this is not a formal proof or an acceptable mathematical argument, but it is good intuition. In order to formalize the argument, recall that we have to set and study whether exists. While we do not have an explicit formula for , we do have a recursive formula, which we recall below.
Since , we can write
Now, if exists and is equal to , we have that as well, so taking the limit of both sides of the above equation gives
This statement is blatantly false, so our underlying assumption that exists is false as well. Since therefore does not exist, must diverge.
As it turns out, the above argument can be used to make a very important observation; if is a sequence for which converges, then . This result is fundamentally important, so we capture it in a theorem.
Stated in plain English, the above test ensures that if the terms in a sequence do not tend to zero, then we cannot add all of the terms in that sequence together.
Let be a sequence and consider the series . If , then we can show that does not exist and hence diverges.
This test gives us a quick way to determine if some series diverge.
In the last example, perhaps the fact that the terms in fluctuate in sign will ensure that the series cannot be infinite. To think about this, let’s turn to the sequence of partial sums. To gain a bit of visual perspective about what is happening, note that the -th term in the sequence of partial sums here is
Plotting several such terms reveals that the terms sequence of partial sums seem to fluctuate.
While we will not show it here, the sequence is bounded; the reason that does not exist is due to the fact that the terms fluctuate (meaning that the sequence is never eventually monotonic).
Let’s summarize the important points from the previous discussion.
- If converges, then .
- If (including the case where the limit does not exist), then diverges.
While divergence test was straightforward to apply in the previous examples, there is a major point to address about what it does not say.
The divergence test can never be used to conclude that a series converges. The theorem does not state that if then converges.
We’ve actually seen an example of this in action.
Note now that the expression in the sum (i.e. the sequence whose terms we are attempting to sum) is , and that since
Thus, we have an example of a sequence whose limit is zero for which the sum of its terms diverges; that is, we have an example where but diverges.
Said another way:
If diverges, it’s still possible that .
Restating this point again (because it is very important): passing the divergence test means that a series has a chance to converge. The divergence test cannot tell us whether a series converges.
There are many questions that require that you now have a firm grasp on the concepts presented thus far. We summarize the important points made thus far, then give many examples that require you to synthesize them.
- There are two fundamental questions we can ask of any sequence.
- Do the numbers in the list approach a finite value?
- Can I sum all of the numbers in the list and obtain a finite result?
These questions can be asked of a given sequence and can also be asked about or any sequence constructed from it.
- Given a sequence , we construct the sequence of partial sum whose -th term is
given by the formula .
- The symbols and are the same.
- By definition converges if exists and in this case, the value of each is the same.
- By definition diverges if does not exist (which includes if the limit is infinte).
- If the limit of a sequence is not zero, the sum of its terms diverges.
- If a series converges, the limit of the sequence whose terms is being summed is zero.
- If the limit of a sequence is zero, more information is needed to determine whether the sum of its terms converges or diverges.
To answer the following questions, make sure that you understand exactly what is given in the statement of the question first, then try to synthesize the material above.
Since , we have diverges by the divergence test.
Note that in the previous questions, was used in two different ways. For the first question, is used to answer a question about by using the definition of convergence. In the second question, we are asked to think of as a sequence in its own right whose terms can be summed. We can use the divergence test to answer this question.
- For the first four choices, notice that since , we have two immediate consequences. First, by definition . Secondly, converges, so .
- Now that we know is a sequence that does not tend to zero, the divergence test tells us we cannot sum its terms; i.e. since , must diverge.
- First, notice that there is a huge difference in the series and , where the latter sum is to be interpreted as . Since , so diverges by divergence test.
- The last choice is never true; we can never determine that a series converges by the divergence test.