Sequences: Basic Definitions

Consider the sequence tex2html_wrap_inline151 . It is clear that we have

displaymath153

The numbers are getting bigger and bigger. Now consider the sequence

tex2html_wrap_inline155
.

In this case, we have

displaymath157

Notice that the numbers are getting smaller and smaller. You may wonder, is it always the case that any sequence of numbers will have one of these two behaviors??? The answer is, of course, NO. Indeed, consider the sequence

displaymath159.

We have

displaymath161.

One may tend to believe that this sequence is getting bigger and bigger. Wrong!!! Indeed, let us go further. We have

displaymath163.

See that the tenth number is equal to the ninth. Nothing wrong yet; however, let us compute the eleventh number

displaymath165.

The tenth number is bigger than the eleventh. The sequence gets bigger and bigger until it reaches the tenth number, then it starts getting smaller and smaller. One may come up with a much simpler example but this example is interesting since most of us would have quit after computing the first five numbers and claim the sequence is getting bigger and bigger!!!!!

Remark: Note that in order to check that the sequence tex2html_wrap_inline167 is getting bigger, we need to check that

displaymath169

This will take a lot of time. There is a shorter way to do that. Indeed, if we check that tex2html_wrap_inline171 for any tex2html_wrap_inline173 , this will be enough. An inductive argument will convince you, I hope....

Definitions: Consider the sequence tex2html_wrap_inline167 . We will say that tex2html_wrap_inline167 is

increasing, if and only if, tex2html_wrap_inline171 for any tex2html_wrap_inline173, or
decreasing, if and only if, tex2html_wrap_inline183 for any tex2html_wrap_inline173.
If one of these properties holds, we say that the sequence is monotonic.

Example: Check that the sequence tex2html_wrap_inline151 is increasing.

Answer: Let tex2html_wrap_inline173 . We have tex2html_wrap_inline191 . Since 2 > 1, then tex2html_wrap_inline195 , which gives

displaymath197.

Example: Check that the sequence tex2html_wrap_inline155 is decreasing.

Answer: Let tex2html_wrap_inline173 . We have n < n+1. Therefore,

displaymath205

holds.

Remark: It may happen sometimes that the sequence tex2html_wrap_inline167 is increasing, if and only if, tex2html_wrap_inline209 for any tex2html_wrap_inline173, and decreasing, if and only if, tex2html_wrap_inline213 for any tex2html_wrap_inline173 . The reader should not panic. No harm will be done.

Now, let us go back to our scientist who collected data every day and where tex2html_wrap_inline217 represents the data collected after n days. What if our scientist discovers that the data of the first seven days are not good? Then he has to throw them out. In this case, he has a new sequence of numbers: tex2html_wrap_inline219 . We will use the notation

displaymath221.

In fact, another way to rewrite this new sequence is

displaymath223.

Clearly, the sequence

displaymath225

represents the case when our scientist throws out the data collected the first k days. This sequence will be referred to as the tail of the original sequence. Also the first k element of the sequence is known as the first block of size k. Using this we see then that the sequence

displaymath159

is decreasing after we throw out the first block of size 9, that is, the tail

displaymath229

is decreasing. Note that we did not check this one before. So it maybe a good idea to train yourself on these kinds of questions.

Remark: Note that there are examples of sequences which do not have a monotonic tail. For example, the sequence tex2html_wrap_inline231 is one of them. It alternates forever between the two numbers 1 and -1.

There is another way of checking whether a sequence has a monotonic tail. This happens whenever the sequence is defined by a function. For example, consider the sequence

displaymath235.

It is not clear or, at least, obvious that this sequence will have a monotonic tail. The reason is that while n grows, the numerator tex2html_wrap_inline239 also grows!! Consider the function

displaymath241,

and compute the derivative

. displaymath243

It is clear that f'(x) < 0, whenever x > e. Since n > e, for any tex2html_wrap_inline251 , then the tail

displaymath253,

is decreasing.

Definitions: Consider the sequence tex2html_wrap_inline167 . We will say that tex2html_wrap_inline167 is

bounded above, if and only if, there exists a number M such that

displaymath261,

for any tex2html_wrap_inline173 . The number M is called an upper-bound.

Furthermore, we will say that tex2html_wrap_inline167 is bounded below, if and only if, there exists a number m such that

displaymath269,

for any tex2html_wrap_inline173 . The number m is called a lower-bound.

If both of these properties hold we say that the sequence is bounded (in short bd).

Example: The sequence tex2html_wrap_inline151 is bounded below by 0 (because it is positive). It is not bounded above.

Example: The sequence tex2html_wrap_inline155 is bounded. Indeed, we have for any tex2html_wrap_inline173 ,

displaymath281.

Therefore, 0 is a lower-bound and 1 is an upper-bound.

Useful Remark: The first block of a sequence is always bounded regardless of its size because we are dealing with finitely many numbers. Therefore, a sequence is bounded (below, above or both), if and only if, one of its tail is bounded (below, above or both).

Next we will discuss the notion of Limit of a sequence.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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