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Online Notes / Calculus II (Notes) / Series & Sequences / Comparison Test/Limit Comparison Test

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In the previous section we saw how to relate a series to an improper integral to determine the convergence of a series.  While the integral test is a nice test, it does force us to do improper integrals which aren’t always easy and in some cases may be impossible to determine the convergence of.

 

For instance consider the following series.

 

 

 

In order to use the Integral Test we would have to integrate

 

 

and I’m not even sure if it’s possible to do this integral.  Nicely enough for us there is another test that we can use on this series that will be much easier to use.

 

First, let’s note that the series terms are positive.  As with the Integral Test that will be important in this section.  Next let’s note that we must have  since we are integrating on the interval .  Likewise, regardless of the value of x we will always have . So, if we drop the x from the denominator the denominator will get smaller and hence the whole fraction will get larger.  So,

 

 

 

Now,

 

 

is a geometric series and we know that since  the series will converge and its value will be,

 

 

 

Now, if we go back to our original series and write down the partial sums we get,

 

 

 

Since all the terms are positive adding a new term will only make the number larger and so the sequence of partial sums must be an increasing sequence.

 

 

 

Then since,

 

 

and because the terms in these two sequences are positive we can also say that,

 

 

 

Therefore, the sequence of partial sums is also a bounded sequence.  Then from the second section on sequences we know that a monotonic and bounded sequence is also convergent.

 

So, the sequence of partial sums of our series is a convergent sequence.  This means that the series itself,

 

is also convergent.

 

So, what did we do here?  We found a series whose terms were always larger than the original series terms and this new series was also convergent.  Then since the original series terms were positive (very important) this meant that the original series was also convergent.

 

To show that a series (with only positive terms) was divergent we could go through a similar argument and find a new divergent series whose terms are always smaller than the original series.  In this case the original series would have to take a value larger than the new series.  However, since the new series is divergent its value will be infinite.  This means that the original series must also be infinite and hence divergent.

 

We can summarize all this in the following test.

 

Comparison Test

Suppose that we have two series  and  with  for all n and  for all n.  Then,

  1. If  is convergent then so is .
  2. If  is divergent then so is .

 

In other words, we have two series of positive terms and the terms of one of the series is always larger than the terms of the other series.   Then if the larger series is convergent the smaller series must also be convergent.  Likewise, if the smaller series is divergent then the larger series must also be divergent.  Note as well that in order to apply this test we need both series to start at the same place.

 

A formal proof of this test is at the end of this section.

 

Do not misuse this test.  Just because the smaller of the two series converges does not say anything about the larger series.  The larger series may still diverge.  Likewise, just because we know that the larger of two series diverges we can’t say that the smaller series will also diverge!  Be very careful in using this test

 

Recall that we had a similar test for improper integrals back when we were looking at integration techniques.  So, if you could use the comparison test for improper integrals you can use the comparison test for series as they are pretty much the same idea.

 

Note as well that the requirement that   and  really only need to be true eventually.  In other words, if a couple of the first terms are negative or  for a couple of the first few terms we’re okay.  As long as we eventually reach a point where   and  for all sufficiently large n the test will work. 

 

To see why this is true let’s suppose that the series start at  and that the conditions of the test are only true for for  and for  at least one of the conditions is not true.  If we then look at  (the same thing could be done for  ) we get,

 

 

 

The first series is nothing more than a finite sum (no matter how large N is) of finite terms and so will be finite.  So the original series will be convergent/divergent only if the second infinite series on the right is convergent/divergent and the test can be done on the second series as it satisfies the conditions of the test.

 

Let’s take a look at some examples.

 

Example 1  Determine if the following series is convergent or divergent.

                                                            

Solution

Since the cosine term in the denominator doesn’t get too large we can assume that the series terms will behave like,

                                                                    

which, as a series, will diverge.  So, from this we can guess that the series will probably diverge and so we’ll need to find a smaller series that will also diverge.

 

Recall that from the comparison test with improper integrals that we determined that we can make a fraction smaller by either making the numerator smaller or the denominator larger.  In this case the two terms in the denominator are both positive. So, if we drop the cosine term we will in fact be making the denominator larger since we will no longer be subtracting off a positive quantity.  Therefore,

 

                                                       

 

Then, since

                                                                     

diverges (it’s harmonic or the p-series test) by the Comparison Test our original series must also diverge.

 

Example 2  Determine if the following series converges or diverges.

                                                                  

Solution

In this case the “+2” and the “+5” don’t really add anything to the series and so the series terms should behave pretty much like

                                                                   

which will converge as a series.  Therefore, we can guess that the original series will converge and we will need to find a larger series which also converges.

 

This means that we’ll either have to make the numerator larger or the denominator smaller.  We can make the denominator smaller by dropping the “+5”.  Doing this gives,

                                                             

 

At this point, notice that we can’t drop the “+2” from the numerator since this would make the term smaller and that’s not what we want.  However, this is actually the furthest that we need to go.  Let’s take a look at the following series.

                                                     

 

As shown, we can write the series as a sum of two series and both of these series are convergent by the p-series test.  Therefore, since each of these series are convergent we know that the sum,

                                                                  

is also a convergent series.  Recall that the sum of two convergent series will also be convergent.

 

Now, since the terms of this series are larger than the terms of the original series we know that the original series must also be convergent by the Comparison Test.

 

The comparison test is a nice test that allows us to do problems that either we couldn’t have done with the integral test or at the best would have been very difficult to do with the integral test.  That doesn’t mean that it doesn’t have problems of its own.

 

Consider the following series.

 

 

This is not much different from the first series that we looked at.  The original series converged because the 3n gets very large very fast and will be significantly larger than the n.  Therefore, the n doesn’t really affect the convergence of the series in that case.  The fact that we are now subtracting the n off instead of adding the n on really shouldn’t change the convergence.  We can say this because the 3n gets very large very fast and the fact that we’re subtracting n off won’t really change the size of this term for all sufficiently large values of n.

 

So, we would expect this series to converge.  However, the comparison test won’t work with this series.  To use the comparison test on this series we would need to find a larger series that we could easily determine the convergence of.  In this case we can’t do what we did with the original series.  If we drop the n we will make the denominator larger (since the n was subtracted off) and so the fraction will get smaller and just like when we looked at the comparison test for improper integrals knowing that the smaller of two series converges does not mean that the larger of the two will also converge.

 

So, we will need something else to do help us determine the convergence of this series.  The following variant of the comparison test will allow us to determine the convergence of this series.

 

Limit Comparison Test

Suppose that we have two series  and  with  for all n.  Define,

                                                                 

If c is positive (i.e.  ) and is finite (i.e.  ) then either both series converge or both series diverge.

 

The proof of this test is at the end of this section.

 

Note that it doesn’t really matter which series term is in the numerator for this test, we could just have easily defined c as,

 

 

and we would get the same results.  To see why this is, consider the following two definitions.

 

 

 

Start with the first definition and rewrite it as follows, then take the limit.

 

 

 

In other words, if c is positive and finite then so is  and if  is positive and finite then so is c.  Likewise if  then  and if  then .  Both definitions will give the same results from the test so don’t worry about which series terms should be in the numerator and which should be in the denominator.  Choose this to make the limit easy to compute.

 

Also, this really is a comparison test in some ways.  If c is positive and finite this is saying that both of the series terms will behave in generally the same fashion and so we can expect the series themselves to also behave in a similar fashion.  If  or  we can’t say this and so the test fails to give any information.

 

The limit in this test will often be written as,

 

 

since often both terms will be fractions and this will make the limit easier to deal with.

 

Let’s see how this test works.

 

Example 3  Determine if the following series converges or diverges.

                                                                  

Solution

To use the limit comparison test we need to find a second series that we can determine the convergence of easily and has what we assume is the same convergence as the given series.  On top of that we will need to choose the new series in such a way as to give us an easy limit to compute for c.

 

We’ve already guessed that this series converges and since it’s vaguely geometric let’s use

                                                                     

as the second series.  We know that this series converges and there is a chance that since both series have the 3n in it the limit won’t be too bad. 

 

Here’s the limit.

                                                            

 

Now, we’ll need to use L’Hospital’s Rule on the second term in order to actually evaluate this limit.

                                                          

 

So, c is positive and finite so by the Comparison Test both series must converge since

 

converges.

 

Example 4  Determine if the following series converges or diverges.

                                                               

Solution

Fractions involving only polynomials or polynomials under radicals will behave in the same way as the largest power of n will behave in the limit.  So, the terms in this series should behave as,

                                                             

and as a series this will diverge by the p-series test.  In fact, this would make a nice choice for our second series in the limit comparison test so let’s use it.

                                             

 

So, c is positive and finite and so both limits will diverge since

                                                                    

diverges.

 

Finally, to see why we need c to be positive and finite (i.e.  and  ) consider the following two series.

 

 

The first diverges and the second converges.

 

Now compute each of the following limits.

 

 

 

In the first case the limit from the limit comparison test yields  and in the second case the limit yields .  Clearly, both series do not have the same convergence.

 

Note however, that just because we get  or  doesn’t mean that the series will have the opposite convergence.  To see this consider the series,

 

 

 

Both of these series converge and here are the two possible limits that the limit comparison test uses.

 

 

 

So, even though both series had the same convergence we got both  and .

 

The point of all of this is to remind us that if we get  or  from the limit comparison test we will know that we have chosen the second series incorrectly and we’ll need to find a different choice in order to get any information about the convergence of the series.

 

We’ll close out this section with proofs of the two tests.

 

Proof of Comparison Test

The test statement did not specify where each series should start.  We only need to require that they start at the same place so to help with the proof we’ll assume that the series start at .  If the series don’t start at  the proof can be redone in exactly the same manner or you could use an index shift to start the series at  and then this proof will apply.

 

We’ll start off with the partial sums of each series.

                                          

 

Let’s notice a couple of nice facts about these two partial sums.  First, because  we know that,

                         

 

So, both partial sums form increasing sequences.

 

Also, because  for all n we know that we must have  for all n.

 

With these preliminary facts out of the way we can proceed with the proof of the test itself.

 

Let’s start out by assuming that  is a convergent series.  Since  we know that,

                                                            

 

However, we also have established that  for all n and so for all n we also have,

                                                                  

 

Finally since  is a convergent series it must have a finite value and so the partial sums,  are bounded above.  Therefore, from the second section on sequences we know that a monotonic and bounded sequence is also convergent and so  is a convergent sequence and so  is convergent.

 

Next, let’s assume that  is divergent.  Because  we then know that we must have  as .  However, we also know that for all n we have  and therefore we also know that  as

 

So,  is a divergent sequence and so  is divergent.

Pf_Box

 

 

Proof of Limit Comparison Test

Because  we can find two positive and finite numbers, m and M, such that .  Now, because  we know that for large enough n the quotient  must be close to c and so there must be a positive integer N such that if  we also have,

                                                                

Multiplying through by  gives,

                                                             

provided .

 

Now, if  diverges then so does  and so since  for all sufficiently large n by the Comparison Test  also diverges.

 

Likewise, if  converges then so does  and since  for all sufficiently large n by the Comparison Test  also converges.

Pf_Box

 


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