Page 91 - Calculus for the Clueless, Calc II

Chapter 14—
Infinite Sequences

This topic brings some controversy. Some people think it is very difficult. Some think it is very easy. I believe
if you understand the beginning, the rest of the chapter is not too bad.

Definition

Sequence—A sequence of terms, technically, is a function for which the domain is the positive integers.
Nontechnically, there is a first term called a 1 (read ''a sub-one," where the "one" is a subscript, not an exponent)
denoting the first term, a2 ("a sub-two") denoting the second term, and so on. The notation for an infinite
sequence is {a n}.

Let us give some examples. We will list some sequences, write the first four terms, and then term number 100
by substituting 1, 2, 3, 4,...,100 for n in a n.
Example 1—

{a n } 1st 2nd 3rd 4th 100th

{6} 6 6 6 6 6

Definition (Nontechnical)

We write if, the larger n gets, the closer an gets to L.

In this case, we say that {a n} converges to L (or has the limit L). If an goes to plus or minus infinity or does not
go to a single number, then {a n} diverges (or has no limit).

Example 2—

Find the limit of {(n + 9)/n }.
2
a n = (n + 9)/n = 1/n + 9/n . As n goes to infinity, both terms go to 0. Therefore, the sequence converges to 0.
2
2
Example 3—

2
Find the limit of {(2n + 3n + 2)/(5 - 7n )}.
2
Divide top and bottom of an by n . We get [2 + (3/n) + (2/n )]/[(5/n -7)]. As n goes to infinity, an goes to 2/(-
2
2
2
7). The sequence has the limit-2/7.
Note

This should look very familiar. This is how we found horizontal asymptotes. Also note that we can use
L'Hopital's rule.