Consider the sequence . It is clear that we have
The numbers are getting bigger and bigger. Now consider the sequence
In this case, we have
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
.
We have
.
One may tend to believe that this sequence is getting bigger and bigger. Wrong!!! Indeed, let us go further. We have
.
See that the tenth number is equal to the ninth. Nothing wrong yet; however, let us compute the eleventh number
.
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 is getting bigger, we need to check that
This will take a lot of time. There is a shorter way to do that. Indeed, if we check that for any , this will be enough. An inductive argument will convince you, I hope....
Definitions: Consider the sequence . We will say that is
Example: Check that the sequence is increasing.
Answer: Let . We have . Since 2 > 1, then , which gives
.
Example: Check that the sequence is decreasing.
Answer: Let . We have n < n+1. Therefore,
holds.
Remark: It may happen sometimes that the sequence is increasing, if and only if, for any , and decreasing, if and only if, for any . 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 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: . We will use the notation
.
In fact, another way to rewrite this new sequence is
.
Clearly, the sequence
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
is decreasing after we throw out the first block of size 9, that is, the tail
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 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
.
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 also grows!! Consider the function
,
and compute the derivative
.
It is clear that f'(x) < 0, whenever x > e. Since n > e, for any , then the tail
,
is decreasing.
Definitions: Consider the sequence . We will say that is
,
for any . The number M is called an upper-bound.
,
for any . The number m is called a lower-bound.
Example: The sequence is bounded below by 0 (because it is positive). It is not bounded above.
Example: The sequence is bounded. Indeed, we have for any ,
.
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