We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.
The reader may notice a connection between this explicit formula and the function whose domain consists of all real nonzero -values.
- and .
- and .
Similarly, if we evaluate the function at for any positive integer , we will find the result matches . Let’s plot the terms in the sequence and the function on the same axes.
The sequence and the function coincide where they are both defined.
A useful first attempt to associate a function of a real variable to a sequence generated by an explicit formula is to try replacing in with and let be the function obtained this way.
Sometimes, we have to work a little harder to find a function of a real variable from which our sequence can be modeled.
Note that while the expressions and may look very different, they generate the same values for each integer . The latter expression, , is necessary to use when finding a function of a real variable that coincides with .
The last example suggests that while there is a connection between sequences and functions of real variables, care must be taken when finding a function of a real-variable that coincides with the sequence on their common domains. Another important point to consider is that many different functions of a real variable can agree when they are evaluated at the integers, as the following example will show.
Let’s see what happens if use a positive integer as an input for each function.
- Since , .
- To evaluate , note that since for all integers . Thus, .
It should not be too surprising that several functions can be used to represent the same sequence because the sequence is defined only for integers, while the functions above are defined for all real -values in their domains.
While there are many examples of sequences, there are two important types of sequences that arise in applications.
The first type of sequence we examine is an arithmetic sequence.
By requiring that the starting index be , the sequence , the terms in the sequence are given explicitly by the formula , or recursively by the rule and . The difference between subsequent elements in this sequence is always four.
In general, the terms in an arithmetic sequence in which subsequent terms differ by can be written as Alternatively, we could describe an arithmetic sequence recursively, by giving a starting value , and using the rule that . You should check that this general statement holds for our two example.
Recall that arithmetic sequences are those where the difference between neighboring elements is constant. Arithmetic sequences are analogues of lines. Consider the earlier example.
By requiring that the lower index of the sequence be , we can check out a graph of the sequence.
By noting that the explicit formula for the -th term in this sequence is , we can try to find a function of a real variable that coincides with this. The simple rule of replacing with works here; we can set and graph both on the same set of axes.
We can graph the terms.
By setting , we can find an explicit formula for the -th term in the sequence; indeed . As such, we can easily find a function of a real variable that coincides with the sequence on their common domains by replacing with to obtain .
We can graph both of these on the same set of axes.
A second very important family of sequences we consider are geometric sequences.
A geometric sequence can also decrease as it progresses.
In general, a geometric sequence in which the ratio between subsequent terms is can be written as Alternatively, we could describe a geometric sequence recursively, by giving a starting value , and using the rule that . As usual, you should check that these general rules hold for the specific examples we’ve considered.
Let’s look at plots of some of our special sequences.
The formula for the terms in the sequence in this example was . We can easily find a function of a real variable that coincides with the sequence on their common domains by replacing with to obtain and graph them both on a common set of axes.
Now let’s consider the second example of a geometric sequence from before, which is illustrated below.
In this example, the explicit formula the terms were given by the formula , so we can find a function of a real variable that coincides with the sequence by setting . We can plot these on the same set of axes.
Can all geometric sequences be modeled using an exponential? A third example explores this. Consider the sequence modeled below.
Replacing with as we did in a previous example gives us a function that agrees with the sequence on their common domains.
Geometric sequences play an important role in the coming sections, so it is useful to have a good feel for what plots of these sequences do!
“Obvious” is the most dangerous word in mathematics” -E.T. Bell