If an infinite sum converges, then its terms must tend to zero.
In order to determine if a series converges, we took the following approach.
- Consider the associated sequence of partial sums, where .
- Try to find an explicit formula for the term . If you can find such a formula, analyze .
- 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 explicitformula for , but unfortunately, this is not always easy or possible. Fortunately, it isnot always necessary to do this in order to determine whether exists. Consider theexample below.
Determine if the series converges or diverges.
Let’s think about what we are trying to do here heuristically. If we set , we can writeout the first several terms in the sequence .
We can observe also that , so eventually, the terms in this list become as close to aswe want. Conceptually, we can interpret that when trying to compute the series, wewill have to add infinitely many numbers together that are very close to .Such an attempt cannot produce a finite result, so we expect the series todiverge.
Of course, this is not a formal proof or an acceptable mathematical argument, but itis good intuition. In order to formalize the argument, recall that we have to set andstudy whether exists. While we do not have an explicit formula for , we do have arecursive 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 sidesof the above equation gives
This statement is blatantly false, so our underlying assumption that exists is false aswell. Since therefore does not exist, must diverge.
As it turns out, the above argument can be used to make a very importantobservation; if is a sequence for which converges, then . This result is fundamentallyimportant, so we capture it in a theorem.
Suppose that is a sequence and the series converges. Then, .
Note that the convergence of the series tells us something about the sequence.
The divergence test
Divergence test Let be a sequence and consider the series . If , then diverges.
Stated in plain English, the above test ensures that if the terms in a sequencedo not tend to zero, then we cannot add all of the terms in that sequencetogether.
The result of this theorem can be established using a similar argument as in theprevious example. Note that there is no explicit reference to the sequence of partialsums in the actual statement of the test. A more complete statement of the testwould be:
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.
Determine if the series converges or diverges.
Here, the sequence whose terms are being summed is given by the formula . Noticethat these terms fluctuate in sign, so maybe when we try adding them all together,we obtain something finite. Let’s try to apply the divergence test. Notice that doesnot exist; it certainly is not 0. Hence, the series diverges by the divergencetest.
In the last example, perhaps the fact that the terms in fluctuate in sign willensure that the series cannot be infinite. To think about this, let’s turn to thesequence of partial sums. To gain a bit of visual perspective about whatis happening, note that the -th term in the sequence of partial sums hereis
Plotting several such terms reveals that the terms sequence of partial sums seem tofluctuate.
While we will not show it here, the sequence is bounded; the reason that does notexist is due to the fact that the terms fluctuate (meaning that the sequence is nevereventually monotonic).
Implications of the divergence test
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 isa 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.
Recall that in a previous section, we showed that the series is actually telescoping.By recognizing that , we showed that there is an explicit formula for the -th term inthe sequence of partial sums given by . We concluded that diverges since.
Note now that the expression in the sum (i.e. the sequence whose terms we areattempting to sum) is , and that since
we have
Thus, we have an example of a sequence whose limit is zero for which the sum of itsterms diverges; that is, we have an example where but diverges.
The standard example of a sequence for which but for which diverges is theharmonic series, . We’ll show later on that this series diverges.
Said another way:
If diverges, it’s still possible that .
To elaborate a little more, we can say that a series “passes the divergence test” if. Which of the following series pass the divergence test?
Restating this point again (because it is very important): passing the divergence testmeans that a series has a chance to converge. The divergence test cannot tell uswhether a series converges.
There are many questions that require that you now have a firm grasp on theconcepts presented thus far. We summarize the important points made thus far, thengive 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 isgiven in the statement of the question first, then try to synthesize the materialabove.
Suppose that is a sequence and let . Suppose that it is also known that.
Which statement below captures the most we can say about ?
converges but we cannot determine its value without more information. converges to . converges to . diverges. More information is needed todetermine if converges.
The information given in the problem is an explicit formula for the terms in thesequence of partial sums , not . To answer this question, we need to know how relates to finding . Since and are analogous representations of the same idea and ,we have converges to .
Which statement below captures the most we can say about ?
converges but we cannot determine its value without more information. converges to . converges to . diverges. More information is needed todetermine if converges.
It may seem daunting to think about what is in general, but note here that theinformation given in the problem is an explicit formula for . is a sequence in its ownright, and we can ask whether we can sum its terms. Here, we can immediately writedown the series in question.
Since , we have diverges by the divergence test.
Note that in the previous questions, was used in two different ways. For the firstquestion, 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 rightwhose terms can be summed. We can use the divergence test to answer thisquestion.
Suppose that is a sequence and let . Suppose that it is known that . What can besaid about ?
converges but we cannot determine its value without more information. converges to . converges to . diverges. More information is needed todetermine if converges.
The information given in the problem is that is a convergent series. We need torelate this to . The relationship between the sequence of partial sums and is that and are analogous, so we need . Thankfully, we have a fact for that; since is convergent, the limit of the inner sequence whose terms we add must be zero; that is .Hence, .
Suppose is a sequence and . Let . Select all statements that must be true:
must diverge. The divergence test tells us converges to.
Let’s look at each of these statements.
- 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.
