Problem: Investigate convergence of the following series. (Does it converge? If it does, then how?)
Solution: We have to check on convergence of the given series and
also on the absolute convergence of it. We start with the former. What can
we say about the terms of this series? It is well-known that
Here in Math Tutor we went a bit further, some advanced courses also mention
less popular but sometimes powerful tests, in particular we introduced
the
Dirichlet test. Can it be used
in our situation? The numbers
Actually, this is a very good question. We already recalled that we usually have three positive and then three negative and then three positive numbers etc. going on, occasionally four of like signs come, so it seems feasible that they would nicely keep cancelling. This is indeed true, but proving it is another matter. A relatively simple argument can be made using complex numbers and a couple of well-known formulas, in particular the formula for partial sums of geometric series.
We have shown that all partial sums are bounded by a common constant, therefore the given series satisfies assumptions of Dirichlet's test and as such it is convergent.
By the way, the number in the denominator above is
Some people may be put off by those complex numbers above and ask: Since the
question has nothing to do with complex numbers, is there a way to prove
this boundedness just using real numbers? Actually, there is, but it is at
least as tricky as the complex numbers approach above. It starts with this
observation. One particular
trig identity can
be used to express
Why on earth would we do such a thing? Notice that we transformed
Uff. I do not know of a nice proof. By the way, we obtained the same upper bound as with the complex numbers approach.
Now we pass to investigation of the absolute convergence of the given series. We are supposed to decide convergence of the series
What test will help us? The sine suggests that we try comparison. We get
Unfortunately, the series on the right is the
harmonic series whose
divergence is well-known, for instance by the
p-test. Thus this
comparison does not help at all. Note that the sine in absolute value
picks values from the
whole range
Since
The lack of convergence of the sine part also rules out limit versions of the Root test and the Ratio test. Since we obviously cannot integrate the sine in absolute value (and the relevant function is not non-increasing anyway), we cannot use the Integral test either and we just ran out of tests to use. Therefore it is definitely worth our time to return to the Root test and Ratio test and inquire about the more general inequality versions of them.
We know from experience that the k in the denominator does not help at all, since in these two tests it gives 1. We can see it when we attempt to use the limit version.
Thus if we want to establish some inequality, we have to use the sine part
for it. And here we are totally out of luck. The sine part is never more
than 1, so we cannot prove divergence in the Root test. On the other hand,
sometimes
We run into similar trouble with the Ratio test. The ratio
We just ran out of tests that people usually know, and also the less popular tests from section More tests in Theory - Testing convergence do not offer any way out.
We are forced to conclude that the tests that we know and standard approaches are not good enough to determine convergence of this series; this means that so far we do not know whether the given series converges absolutely.
Note that if we did not have the Dirichlet test, we would have known absolutely nothing about our series. The Alternating series test could not be used; in such a case we typically pass to absolute convergence, but here we also ran afoul of troubles. So the tools that are traditionally covered in calculus courses are powerless in this problem. There actually is a way to decide convergence, but it requires that we look closer at the given series.
It is essentially the
harmonic series whose
terms were modified. What happens to them? We know that for infinitely many
k the numbers
A good start is observation that for every small value of
There are at least two approaches to do this, two possibilities to get a more
tangible observation. First, note that the couple k,
It seems that at least one end always yields large value of sine, sometimes
both. Since we do not know which ends is the large one, the best way to
express this is to actually talk about the sum of the two values. We claim
that there is a certain positive number a such that the sum of
Having this proved we can show that our series diverges. First we group its terms into couples.
Since all the terms in this series are positive, the associative law can be applied here, (this is actually not so obvious as it seems, see Basic properties in Theory - Introduction). Thus convergence of the series on the left is equivalent to convergence of the new series on the right. However, there we can use comparison.
This proves that the series on the left is divergent, so the series with
absolute values is divergent and therefore the given series is not
absolutely convergent.
Conclusion: The given series converges conditionally.
Alternative: When imagining what can happen on the graph of sine we
may notice also this: When we take three successive integers, then at least
one of them makes
Having this established we proceed similarly as before. First we group our series into triplets.
Now we look at one such triplet, for simplicity we put some positive
constants on the top and using our observation we know that at least one of
them is greater than
Using this in the decomposition of our series into triplets we obtain
This confirms divergence of our series.
So we won in the end, but it was tough work. Note that there are series that beat even the most advanced tests and tricks, for instance convergence or divergence of the following series, although it is rather nice, is still unknown.
(At least it was in 2004 when I last checked, so even if it were not true now, it shows that a simple series like this can withstand several centuries of attacks by all kinds of convergence tests and tricks.)