Introduction to Sequences & Series
A mathematical sequence is an ordered list of objects, often numbers. Sometimes the numbers are defined in terms of a previous number in the list.
Key Points
- The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.
- An arithmetic sequence is one in which a term is obtained by adding a constant to a previous term of a sequence. So, the $n^{th}$ term can be described by the formula $ a_{n}=a_{n-1}+d $
- A geometric sequence is one in which a term of a sequence is obtained by multiplying the previous term by a constant. It can be described by the formula $ a_{n}=r(a_{n-1}) $
1. Sequences: Recursive and Explicit
DEFINTIION
A finite sequence with terms in a set $S$ is a function on the natural numbers (i.e. the positive integers) from ${1, 2, …, n}$ to $S$ for some $n>0$. An infinite sequence, $S$, is a function that continues on the natural numbers indefinately from $\mathbb{N}$ to $S$. For example, the sequence of prime numbers (2, 3, 5, 7, 11, … ) is the function $$1\to 2,2\to 3,3\to5,4\to7,5\to11,6$$
More simply put, a sequence is an ordered list of objects. Like a set, it contains terms (also called elements or members). The number of ordered elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and a particular term can appear multiple times at different positions in the sequence.
For example, (M,A,R,Y) is a sequence of letters that differs from (A,R,M,Y), as the ordering matters, and (1,1,2,3,5,8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,⋯). Finite sequences are sometimes known as “strings” or “words” and infinite sequences as “streams.”
If you disregard the very simplest cases, there is in all of mathematics not a single infinite series whose sum has been rigorously determined. In other words, the most important parts of mathematics stand without a foundation.
Abel, Niels H. (1802 – 1829) In G. F. Simmons, Calculus Gems, New York: McGraw Hill, Inc., 1992, p. 188.
Sequence Notation:
With sequence notation, we typically do not use traditional function notation. Rather, the sequence definition is given by $a_n=$, not $f(x)=$. Additionally, we use subscripted letters appended to letters like $a_1$ to represent the terms of the sequence and their location withing the sequence. Sometimes a sequence may begin on $a_0$ or $a_1$ depending one whether the sequence is defined to to include or not include $0$ in its domain.
When we have a formula that defines the $n^{th}$ term of a sequence ($a_n$ or the general term), we ususally reprent the entire sequence by placing braces around the formula.
For example, tthe sequence whose $n^{th}$ term is $b_n=\frac{1+n}{2^n}$ may be represented as
$$\{b_n\}=\{\frac{1+n}{2^n}\}$$
or by
$b_1=1$, $b_2=\frac{3}{4}$, $b_3=\frac{4}{9}$, $b_4=\frac{5}{16}$, … , $b_n=\frac{n+1}{2^n}$, …
Since we have the formula for the general term of the sequence, we can find any term by substituting that value for $n$ into the sequence definition. For example, the 21st term of this sequence above is
$$b_n=\frac{n+1}{2^n}=\frac{21+1}{2^21}=\frac{22}{2,097,152}=\frac{11}{1,048,576}$$
Graphing provides a visual representation of the sequence as a set of distinct points. We can see from the appended graph in Figure 1 that the terms are converging towards teh x-axis at an exponential rate. This particular sequence forms an exponential decay function.
Often, we might need to determine the general form of a sequence simply by observing a pattern in the terms themselves. This can often be quite difficult. In this chapter, we will examine a variety of methods to generate general forms for various kinds of sequences, though often there is no clear method other than mere trial-and-error and careful observation of how the terms change according to the value of $n$ (i.e. their term number in the sequence). Consider some of the below examples and try to determine the pattern definition yourself.
1. $e, \frac{e^2}{2}, \frac{e^3}{3}, \frac{e^4}{4}, …$ (click to see solution)
$a_n=\frac{e^n}{n}$
2. $1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, …$ (click to see solution)
$b_n=\frac{1}{4^{n-1}}$
3. $1, 3, 5, 7, 11, …$ (click to see solution)
$c_n=2n-1$
4. $1, 4, 9, 16, 25, …$ (click to see solution)
$d_n=n^2$
5. $1, 3, 8, 15, 24, …$ (click to see solution)
$e_n=n^2-1$
6. $1, \frac{-1}{2}, \frac{1}{3}, \frac{-1}{4}, \frac{1}{5}, …$ (click to see solution)
$f_n=\frac{(-1)^{n-1}}{n}$
Recursive Sequences:
Many of the sequences you will encounter in mathematics are produced by a formula, where some operation(s) is performed on the previous member of the sequence $a_{n-1}$ to give the next member of the sequence $a_{n}$. These are called recursive sequences.
Arithmetic Sequences:
An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term. An example is (10,13,16,19,22,25). In this example, the first term (which we will call $a_{1}$) is 10, and the common difference ($d$) — that is, the difference between any two adjacent numbers — is 3. The recursive definition is therefore: $$a_{n}=a_{n-1}+3, \quad a_{1}=10$$
Another example is (25,22,19,16,13,10). In this example $a_{1}=25$, and $d=-3$. The recursive definition is therefore $$a_{n}=a_{n-1}-3, \quad a_{1}=25$$
In both of these examples, $n$ (the number of terms) is 6.
Geometric Sequences:
A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number. An example is (2,6,18,54,162). In this example, $a_{1}=2$, and the common ratio $r$ — that is, the ratio between any two adjacent numbers — is 3. Therefore, the recursive definition is: $$a_{n}=3(a_{n-1}), \quad a_{1}=2$$
Another example is (162,54,18,6,2). In this example $a_{1}=162$, and $r=\frac{1}{3}$. Therefore, the recursive formula is: $$a_{n}=\frac{1}{3}(a_{n-1}), \quad a_{1}=162$$
In both examples $n=5$.
Recursive Definitions:
Divergent Sequence:
Some sequences “diverge” and go to infinity (positive and/or negative) as n becomes infinitely larger:
$$[a_{n}]{1}^{\infty}=3n+2^{n} \newline$$
$$a{1}=3(1)+2^{1}=5 \newline$$
$$a_{2}=3(2)+2^{2}=10 \newline$$
$$a_{3}=3(3)+2^{3}=17 \newline$$
$$a_{4}=3(4)+2^{4}=28 \newline$$
$$a_{5}=3(5)+2^{5}=47 \newline$$
$$a_{5}=3(6)+2^{6}=82 \newline$$
Convergent Sequences:
Some sequences “converge” and go to some infinitesimally precise value as n becomes infinitely larger:
$$[a_{n}]{1}^{\infty}=\frac{1}{n^{2}} \newline$$
$$a{1}=\frac{1}{1^{2}}=1 \newline$$
$$a_{2}=\frac{1}{2^{2}}=\frac{1}{4} \newline$$
$$a_{3}=\frac{1}{3^{2}}=\frac{1}{9} \newline$$
$$a_{4}=\frac{1}{4^{2}}=\frac{1}{16} \newline$$
$$a_{5}=\frac{1}{5^{2}}=\frac{1}{25} \newline$$
$$a_{6}=\frac{1}{6^{2}}=\frac{1}{36} \newline$$
Alternating Sequences:
Some sequences “alternate,” oscillating between two values, one the “upper bound” and the other the “lower bound”:
$$[a_{n}]{1}^{\infty}=3+(-1)^{n}\newline$$
$$a{1}=3+(-1)^{1}=2\newline$$
$$a_{2}=3+(-1)^{2}=4\newline$$
$$a_{3}=3+(-1)^{3}=2\newline$$
$$a_{4}=3+(-1)^{4}=4\newline$$
$$a_{5}=3+(-1)^{5}=2\newline$$
$$a_{6}=3+(-1)^{6}=4\newline$$
An Interesting Sequence: The Fibonacci
The Fibonacci Sequence:
The Fibonacci Sequence begins as 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. …
This can be written using a recurrsive sequence as given below:
$$a_{k}=a_{k-2)+a_{k-1}$$
where $a_0=1$ and $a_1=1$.
$$a_{2}=a_{0)+a_{1}=1+1=2$$
$$a_{3}=a_{1)+a_{2}=1+2=3$$
$$a_{4}=a_{2)+a_{3}=2+3=5$$
and so on.
How is the Fibonacci Sequence associated with the Golden Ratio? We merely take the sequence as a sequence of progressive quotients and observe its behavior at very large values on $n$ approaching infinity.
$${{F_k}_0^{\infty}}=\frac{a_k}{a_{k-1}}$$
where $a_0=1$ and $a_1=1$ and $a_k=a_{k-2}+a_{k-1}$
$F_1=\frac{a_1}{a_0}=\frac{1}{1}=1$ and $a_2=a_0+a_1=1+1=2$
$F_2=\frac{a_2}{a_1}=\frac{2}{1}=2$ and $a_3=a_1+a_2=1+2=3$
$F_3=\frac{a_3}{a_2}=\frac{a_1+a_2}{a_2}=\frac{3}{2}$ and $a_4=a_3+a_2=3+2=5$
$F_4=\frac{a_4}{a_3}=\frac{a_2+a_3}{a_3}=\frac{5}{3}$ and $a_5=a_4+a_3=5+3=8$
$F_5=\frac{a_5}{a_4}=\frac{a_3+a_4}{a_4}=\frac{8}{5}$ and $a_6=a_5+a_4=8+5=13$
$F_6=\frac{a_6}{a_5}=\frac{a_4+a_5}{a_5}=\frac{13}{8}$ and $a_7=a_6+a_5=13+8=21$
and so on. Thus, we have the sequence of quotients:
$$\frac{1}{1},\frac{2}{1},\frac{3}{2},\frac{5}{3},\frac{8}{5},\frac{13}{8},\frac{21}{13},…\text{etc.}$$
These values are remarkably approaching $\frac{1+\sqrt{5}}{2}$ at the limit of infinity. This is the Folden Ratio which is approximately 1.6180… .
Thus, while the Fibonacci Sequence is a divergent sequence, the sequence of quotients made from the terms of the Fibonacci Sequence is a convergent sequence.
An explicit definition of an arithmetic sequence is one in which the $n^{th}$ term is defined without referring to the previous term. This is more useful, because it means you can find (for instance) the 20th term without finding all of the other terms in between.
To find the explicit definition of an arithmetic sequence, you begin writing out the terms. Assume our sequence is $t_{1},t_{2},…$. The first term is always $t_{1}$. The second term goes up by $d$, and so it is $t_{1}+d$. The third term goes up by $d$ again, and so it is $(t_{1}+d+d$, or in other words, $t_{1}+2d$. So we see that:
$$t_{1}=t_{1}\newline$$
$$t_{2}=t_{1}+d\newline$$
$$t_{3}=t_{1}+2d\newline$$
$$t_{4}=t_{1}+3d\newline$$
$$…$$
and so on. From this we can see the generalization that:
$$t_{n}=t_{1}+(n-1)d$$
which is the explicit definition we were looking for.
The explicit definition of a geometric sequence is obtained in a similar way. The first term is $t_{1}$; the second term is $r$ times that, or $t_{1}r$; the third term is $r$ times that, or $t_{1}r^{2}$ ; and so on.
$$t_1={t_1}r^0$$
$$t_2={t_1}r^1$$
$$t_3={t_2}r=({t_1}r^1)r={t_1}r^2$$
$$t_4={t_3}r=({t_1}r^2)r={t_1}r^3$$
$$t_5={t_4}r=({t_1}r^3)r={t_1}r^4$$
$$…$$
$$t_n=t_{n-1}r=({t_1}r^{n-2})r={t_1}r^{n-2+1}={t_1}r^{n-1}$$
So, the general rule is:
$$t_{n}=t_{1}r^{n-1}$$
which is the explicit definition we were looking for.
2. The Explicit Form of a Polynomial Sequence
Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence. Such a formula will produce the $n^{th}$ term when a value for the integer $n$ is put into the formula.
If a sequence is generated by a polynomial, this fact can be detected by noticing whether the computed differences eventually become constant.
Linear Polynomials:
Consider the sequence: 5,7,9,11,13,…
The difference between 7 and 5 is 2. The difference between 7 and 9 is also 2. In fact, the difference between each pair of terms is 2. Since this difference is constant, and this is the first set of differences, the sequence is given by a first-degree (linear) polynomial.
Suppose the formula for the sequence is given by $an+b$ for some constants $a$ and $b$. Then the sequence looks like:
$$a+b \quad , \quad 2a+b \quad , \quad 3a+b \quad , \quad …
The difference between each term and the term after it is $a$. In our example, $a=2$. It is possible to solve for $b$ using one of the terms in the sequence. Using the first number in the sequence and the first term:
$$5+a+b \newline$$
$$b=5-a \newline$$
$$b=5-(2) \newline$$
$$b=3 \newline$$
So, the $n^{th}$ term of the sequence is given by $2n+3$.
Quadratic Polynomials:
Suppose the $n^{th}$ term of a sequence was given by $an^{2}+bn+c$. Then the sequence would look like:
$$a+b+c , 4a+2b+c , 9a+3b+c , …$$
This sequence was created by plugging in 1 for $n$, 2 for $n$, 3 for $n$, etc.
If we start at the second term and subtract the previous term from each term in the sequence, we can get a new sequence made up of the differences between terms. The first sequence of differences would be:
$$2a+b , 5a+b , 7a+b , …$$
Now, we take the differences between terms in the new sequence. The second sequence of differences is:
$$2a , 2a , 2a , 2a , …$$
The computed differences have converged to a constant after the second sequence of differences. This means that it was a second-order (quadratic) sequence. Working backward from this, we could find the general term for any quadratic sequence.
Example: Consider the sequence: 4,−7,−26,−53,−88,−131,…
The difference between −7 and 4 is −11, and the difference between −26 and −7 is −19. Finding all these differences, we get a new sequence:
$$-11,−19,−27,−35,−43,…$$
This list is still not constant. However, finding the difference between terms once more, we get:
$$-8,−8,−8,−8,…$$
This fact tells us that there is a polynomial formula describing our sequence. Since we had to do differences twice, it is a second-degree (quadratic) polynomial.
We can find the formula by realizing that the constant term is −8, and that it can also be expressed by $2a$. Therefore $a=−4$.
Next we note that the first item in our first list of differences is −11, but that generically it is supposed to be $3a+b$, so we must have $3(-4)+b=-11$, and $b=1$.
Finally, note that the first term in the sequence is 4, and can also be expressed by
$$a+b+c=-4+1+c$$
So, $c=7$, and the formula that generates the sequence is
$$-4a^{2}+b+7c$$
General Polynomial Sequences
This method of finding differences can be extended to find the general term of a polynomial sequence of any order. For higher orders, it will take more rounds of taking differences for the differences to become constant, and more back-substitution will be necessary in order to solve for the general term.
General Terms of Non-Polynomial Sequences
Some sequences are generated by a general term which is not a polynomial. For example, the geometric sequence 2, 4, 8, 16, … is given by the general term $2n$. Because this term is not a polynomial, taking differences will never result in a constant difference.
General terms of non-polynomial sequences can be found by observation, as above, or by other means which are beyond our scope for now. Given any general term, the sequence can be generated by plugging in successive values of $n$.
3. Series and Sigma Notation
Sigma notation, denoted by the uppercase Greek letter sigma (Σ), is used to represent summations—a series of numbers to be added together.
Summation is the operation of adding a sequence of numbers, resulting in a sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum. The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. For finite sequences of such elements, summation always produces a well-defined sum.
A series is a list of numbers—like a sequence—but instead of just listing them, the plus signs indicate that they should be added up. For example, 4+9+3+2+17 is a series. This particular series adds up to 35. Another series is 2+4+8+16+32+64. This series sums to 126.
Sigma Notation
One way to compactly represent a series is with sigma notation, or summation notation, which looks like this:
$$\sum_{n=3}^{7}n^{2}$$
The main symbol seen is the uppercase Greek letter sigma. It indicates a series. To “unpack” this notation, $n=3$ represents the number at which to start counting (3), and the 7 represents the point at which to stop. For each term, plug that value of $n$ into the given formula ($n^{2}$). This particular expression, which we can read as “the sum as $n$ goes from 3 to 7 of $n^{2}$,” means:
$$3^{2}+4^{2}+5^{2}+6^{2}+7^{2}$$
More generally, sigma notation can be defined as:
$$\sum_{i=m}^{n}x_{i}=x_{m}+x_{m+1}+x_{m+2}+…+x_{n-1}+x_{n}$$
In this formula, $i$ represents the index of summation, $x_{i}$ is an indexed variable representing each successive term in the series, $m$is the lower bound of summation, and $n$ is the upper bound of summation. The “$i=m$” under the summation symbol means that the index $i$ starts out equal to $m$. The index, $i$, is incrememnted by 1 for each successive term, stopping when $i=n$. It is worth noting that sometimes the letter $k$ is used instead of $i$ ; herein, we will use both interchangeably.
Informal writing sometimes omits the definition of the index and bounds of summation when these are clear from context. For example:
$$\sum x_{i}^{2}=\sum_{i=1}^{n}x_{i}^{2}$$