MATH 155 SUPPLEMENTAL NOTES 19

LIMITS OF SEQUENCES

In the previous set of supplemental notes, I defined what is a sequence and started to talk about the convergence or divergence of a sequence. In that set of notes, the approach that was taken was graphical, and as you could see, it was not the most efficient way to determine the convergence. In this set of notes, we will discuss different ways of determining the convergence or divergence of a sequence.

First of all, let us discuss the idea of using a limit to determine the convergence or divergence of a sequence.

 FACT: 1. Sum Rule 2. Difference Rule 3. Product Rule 4. Constant Multiple Rule 5. Quotient Rule

 EXAMPLE 1: Does the following sequence converge or diverge? SOLUTION: I will first use long division to simplify this rational expression. Now I will look at the limit. Since the limit approaches a finite value, we can conclude that the sequence converges.

 EXAMPLE 2: Does the following sequence converge or diverge? SOLUTION: Again, I will use long division to simplify this radical expression. Now, I will look at the limit of this sequence. Since the limit exists, we can conclude that the sequence converges.

 EXAMPLE 3: Does the following sequence converge or diverge? SOLUTION: First let us simplify this rational expression. Now, determine the limit of this sequence. Since the limit goes off to infinity, therefore, the sequence diverges.

 EXAMPLE 4: Does the following sequence converge or diverge? SOLUTION: Therefore, this sequence does converge.

 EXAMPLE 5: Does the following sequence converge or diverge? SOLUTION: How are we going to deal with the fact that we have a cosine in this sequence? Well, if you can remember when we discussed limits in Calculus I, you should remember that there is a theorem called the Sandwich Theorem that we used for functions similar to this function. Here is the Sandwich Theorem. FACT: We now have to determine if the two new sequences converge or diverge to the same value. Since both limits converge to the same value, therefore the limit of a n also converges to 0 by the sandwich theorem.

 EXAMPLE 6: Does the following sequence converge or diverge? SOLUTION: We know that -1 £ sin n £ 1, which implies 0 £ sin 2 n £ 1, therefore Therefore, by the sandwich theorem, the sequence a n converges.

 For some of these problems, especially the first few that were discussed, it would have been nice to be able to use l ' Hôpital's Rule, but we could not because we are talking about sequences and not functions. The next two facts will help us make a connection between sequences and functions, so we will be able to use l ' Hôpital's Rule.

 FACT: The Continuous Function Theorem for Sequences Let {a n} be a sequence of real numbers. If a n ® L and if f is a function that is continuous at L and defined at all a n, then f (a n) ® f (L).

 EXAMPLE 7: Does the following sequence converge or diverge? SOLUTION: How did I simplify b n? I used long division to simplify b n. Now let us determine if the sequence b n converges or diverges.

 Here is the second fact. FACT: Suppose that f (x) is a function defined for all x ³ n 0 and that {a n} is a sequence of real numbers such that a n = f (n) for n ³ n 0. This fact is stating that the corresponding function f (x) will have the same limit as its corresponding sequence, a n.

 EXAMPLE 8: Does the following sequence converge or diverge? SOLUTION: So let us look at the limit of f (x). (The first limit evaluates to the indeterminate form ¥ / ¥.) Since the limit exists, we can conclude that a n converges to zero. Isn't it nice when we can use l ' Hôpital's Rule!

 EXAMPLE 9: Does the following sequence converge or diverge? SOLUTION: Since the limit exists, we can conclude that the sequence a n converges to zero.

 EXAMPLE 10: Does the following sequence converge or diverge? SOLUTION: As you learned in supplemental notes 6, ¥ - ¥ is an indeterminate form. I am going to convert this indeterminate form into the indeterminate form of ¥ / ¥ by rationalizing the numerator. This is now in the form of ¥ / ¥. Now, I am going to divide both top and bottom by x, taking note to divide everything inside the radical by x 2. Therefore the sequence a n converges.

 The more that you do of these types of problems, you will find out that there are some limits that occur frequently.  Hence it might be easier to remember what form the limit will take. Here are the limits that you should know For equations 3 - 6, x is number that remains fixed.

 EXAMPLE 11: Does the following sequence converge or diverge? SOLUTION:

 EXAMPLE 12: Does the following sequence converge or diverge? SOLUTION:

 EXAMPLE 13: Does the following sequence converge or diverge? SOLUTION:

 EXAMPLE 14: Does this sequence converge or diverge? SOLUTION:

 EXAMPLE 15: Does the following sequence converge or diverge? SOLUTION: So this is in the form of equation 2.

As you can see, limits of sequences can be treated like the limit of functions. Therefore we can use the methods that we learned in calculus I and II. Work through these examples taking note of the six frequently seen limits. As with any of these notes, if you have any questions on any of the examples worked in these notes, please feel free to contact me.