Limits of Infinite Sequences

13 - 4 Limits of Infinite Sequences

Important stuff coming!

A series that does not have a last term is called infinite. Look at the following sequence: 1/2, 1/4, 1/8, 1/16, . . . (1/2)^x . . . How small do these terms get? Use your calculator and put in some large values for x and see what happens. Obviously, the numbers are getting smaller. Is there a number that the sequence approaches? Take a look at the graph of this sequence.

You can see that the graph is asymptotically approaching the x-axis. This means that the sequence approaches the number zero!! You have just found your first limit. (Hope we are not "limiting" ourself too much!!)

Conclusion:

(1/2)^x = 0

We have seen an example of a sequence approaching a number. What other possibilities await for us?

1) The sequence can approach a number as we have just seen.

2) The sequence can approach one of the two infinities. Either positive or negative infinity. Example : 1, 2, 4, 8, 16, . . . approaches positive infinity, while the sequence : -1, -2, -4, -8, -16, . . . approaches negative infinity.

3) The sequence has no limit. The big deal here is that the limit has to approach a single value or go towards one of the infinities. (Note: infinity is not a number. It means that the sequence increases without bound!) Example: 1, -2, 3, 4, 5, -6, . . . doesn't approach one single value. The purple odd numbers 1, 3, 5, 7, . . . go toward positive infinity, while the pink even numbers -2, -4, -6, . . . go toward negative infinity. Result: since they don't approach the same thing, there is no limit.

Study this example closely as it will lead to an important theorem!! (I'm not kidding!!)

(0.99)^x = ? Use your calculator, and plug in increasing values for x. Do you see that as you use higher and higher values, the sequence approaches zero!! Notice I took a number fairly close to 1. If you replace any fraction between 0 and 1, won't the limit also approach zero? This leads us to an important theorem and a method to calculate limits!

Theorem

If |r| < 1, then r^x = 0

This theorem implies that any fraction having x to a positive power in the denominator (power in the denominator > power of x in the numerator) will approach zero!!

(1/x) = 0 or

(1/x²) = 0, etc. This gives us a way to calculate some limits. If the problem is a fraction, try dividing the numerator and denominator by the highest power of x in the denominator!

Example:

To find the limit, divide top and bottom by x²

Simplifying yields -1.

Notice that both (3/x²) and (2/x²) approach zero because of the previous theorem!!

Other Limits:

Find cos(1/x)

Solution: Because the limit of 1/x = 0 because of the theorem, we are basically asking, what is the cos 0? How about 1.

Therefore, the limit is 1.

Find

(1.35)^x

Solution: Since 1.35 is larger than one, the theorem dosesn't work. As you take higher powers of x, the answers keep increasing unbounded.

Therefore, the limit is positive infinity.

Find the limit of the sequence 1, 2, 1, 2, 1, 2 ,1 ,2 . . .

Solution: Look at the pattern. Basically, it's two patterns, alternating between 1 and 2. Since the first approaches 1 and the second approaches 2, it is approaching two values.

Therefore, there is no limit. (Remember the 3 possibilities at the beginning?)

Find the limit of the sequence: 1/2, -1/3, 1/4, -1/5, 1/6, -1/7 . . .

Solution: Again, split it into two patterns

1/2, 1/4, 1/6, 1/8, . . . This approaches 0.

-1/3, -1/5, -1/7, -1/9, . . . This also approaches 0.

Therefore, the limit is 0 because they both approach the same number.

That's a basic look at some limits of sequences. The next section deals with how to sum an infinite series. Hope this hasn't been a delimiting experience for you, but everything should be "OK"!

Pooh says, "On to the series sums and we won't take that page to "serious"!