What is a sequence?

A sequence is an infinitely long list of numbers.

Take a look at this sequence of fractions:

1/21 / 2
2/32 / 3
3/43 / 4
4/54 / 5
5/65 / 6
6/76 / 7
7/87 / 8
8/98 / 9
9/109 / 10
10/1110 / 11
11/1211 / 12
12/1312 / 13
13/1413 / 14
14/1514 / 15
15/1615 / 16
16/1716 / 17
17/1817 / 18
18/1918 / 19
19/2019 / 20
20/2120 / 21
21/2221 / 22
22/2322 / 23
23/2423 / 24
24/2524 / 25
25/2625 / 26
26/2726 / 27
27/2827 / 28
28/2928 / 29
29/3029 / 30
30/3130 / 31
31/3231 / 32
32/3332 / 33
33/3433 / 34
34/3534 / 35
35/3635 / 36
36/3736 / 37
37/3837 / 38
38/3938 / 39
39/4039 / 40
40/4140 / 41
41/4241 / 42
42/4342 / 43
43/4443 / 44
44/4544 / 45
45/4645 / 46
46/4746 / 47
47/4847 / 48
48/4948 / 49
49/5049 / 50
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Notice how the sequence is infinitely long (you can keep scrolling forever) and each term of the sequence is labeled with a number.

These numbers, starting with 1 and counting up, are called the β€œnatural numbers”, sometimes indicated by the symbol N\mathbb{N}:

N={1,2,3,4,5,6,7,8,9,10,...}\mathbb{N} = \{ \textcolor{#1d4ed8}{1, 2, 3, 4, 5, 6, 7, 8, 9, 10,...} \}

Since each natural number corresponds with a term in the pizza sequence, we say that this sequence is indexed by the natural numbers.

Side note

It's also possible to have a sequence that is indexed by some other set, like the integers: Z={...,βˆ’2,βˆ’1,0,1,2,...}\mathbb{Z} = \{ ..., -2, -1, 0, 1, 2,... \}. That kind of sequence would be infinite in both directions, not just to the right.

Notation for Sequences

A sequence is very similar to a function. For example, our pizza sequence is a little bit like the function f(n)=nn+1f(\textcolor{#1d4ed8}{n}) = \frac{n}{n + 1}. (Becausef(1)=12f(\textcolor{#1d4ed8}{1}) = \frac 12, f(2)=23f(\textcolor{#1d4ed8}{2}) = \frac 23, and so on.)

But we don't generally use function notation for sequences. Instead, we write them like this:(nn+1)n∈N\left(\frac{n}{n+1}\right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}This notation basically means "compute nn+1\frac{n}{n + 1} for all nn in N\mathbb{N}". Which makes sense, because that's exactly what our pizza sequence is.

But sometimes we don't know exactly what our sequence is. When writing proofs, we will often discuss mystery sequences, where we know some properties of the sequence but not its exact values. In that case, we can't write an exact formula for the terms, so we just write(xn)n∈N(x_n)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}instead. Here, xnx_n is a mystery function that has an unknown value for each n\textcolor{#1d4ed8}{n}.

Side note

If you're in the mood for using mystery notation, you could describe the pizza sequence as "(xn)n∈N(x_n)_{n \in \mathbb{N}} where xn=nn+1x_n = \frac{n}{n+1} for all nn in N\mathbb{N}". Of course, describing the values of xnx_n really ruins the mystery.

Graphing Sequences

It is much easier to understand the behavior of a sequence if you have a graph:

0.40.50.60.70.80.911.1
nn+1\frac{n}{n+1}
1/21 / 2
2/32 / 3
3/43 / 4
4/54 / 5
5/65 / 6
6/76 / 7
7/87 / 8
8/98 / 9
9/109 / 10
10/1110 / 11
11/1211 / 12
12/1312 / 13
13/1413 / 14
14/1514 / 15
15/1615 / 16
16/1716 / 17
17/1817 / 18
18/1918 / 19
19/2019 / 20
20/2120 / 21
21/2221 / 22
22/2322 / 23
23/2423 / 24
24/2524 / 25
25/2625 / 26
26/2726 / 27
27/2827 / 28
28/2928 / 29
29/3029 / 30
30/3130 / 31
31/3231 / 32
32/3332 / 33
33/3433 / 34
34/3534 / 35
35/3635 / 36
36/3736 / 37
37/3837 / 38
38/3938 / 39
39/4039 / 40
40/4140 / 41
41/4241 / 42
42/4342 / 43
43/4443 / 44
44/4544 / 45
45/4645 / 46
46/4746 / 47
47/4847 / 48
48/4948 / 49
49/5049 / 50
n∈N\textcolor{blue}{n \in \mathbb{N}}
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(nn+1)n∈N\left( \frac{n}{n+1} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

It looks like the terms of this sequence get closer and closer to 1 as n\textcolor{#1d4ed8}{n} gets larger. This concept of converging to a value is essential in analysis, so keep your eyes peeled for graphs that look like this.

Side note

It makes sense that the sequence converges to 1, because for large n\textcolor{#1d4ed8}{n}, the terms become fractions like 999,9991,000,000\frac{999,999}{1,000,000}, which is pretty much 1.

Sequence Gallery

The best way to become familiar with sequences is to look at a lot of them. The following gallery contains a selection of interesting sequences:

-60-40-200204060
(n)n∈N\left( n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}
-101
(1n)n∈N\left( \frac{1}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}
-101
((βˆ’1)n)n∈N\left( (-1)^n \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}
-101
((βˆ’1)nn)n∈N\left( \frac{(-1)^n}{n} \right)_{\textcolor{#1d4ed8}{n \in \mathbb{N}}}

If you had to classify these sequences into categories, how might you divide them up? (In the next section, we will discuss one way that mathematicians classify sequences.)

Build Your Own Sequence

Of course! I can't show you all these examples without giving you the chance to build a sequence of your own.

Enter an expression using n\textcolor{#1d4ed8}{n} into the box below to create a sequence.

0.40.50.60.70.80.91
nn+1\frac{n}{n+1}
12\frac{1}{2}
23\frac{2}{3}
34\frac{3}{4}
45\frac{4}{5}
56\frac{5}{6}
67\frac{6}{7}
78\frac{7}{8}
89\frac{8}{9}
910\frac{9}{10}
1011\frac{10}{11}
1112\frac{11}{12}
1213\frac{12}{13}
1314\frac{13}{14}
1415\frac{14}{15}
1516\frac{15}{16}
1617\frac{16}{17}
1718\frac{17}{18}
1819\frac{18}{19}
1920\frac{19}{20}
2021\frac{20}{21}
2122\frac{21}{22}
2223\frac{22}{23}
2324\frac{23}{24}
2425\frac{24}{25}
2526\frac{25}{26}
2627\frac{26}{27}
2728\frac{27}{28}
2829\frac{28}{29}
2930\frac{29}{30}
3031\frac{30}{31}
3132\frac{31}{32}
3233\frac{32}{33}
3334\frac{33}{34}
3435\frac{34}{35}
3536\frac{35}{36}
3637\frac{36}{37}
3738\frac{37}{38}
3839\frac{38}{39}
3940\frac{39}{40}
4041\frac{40}{41}
4142\frac{41}{42}
4243\frac{42}{43}
4344\frac{43}{44}
4445\frac{44}{45}
4546\frac{45}{46}
4647\frac{46}{47}
4748\frac{47}{48}
4849\frac{48}{49}
4950\frac{49}{50}
5051\frac{50}{51}
5152\frac{51}{52}
5253\frac{52}{53}
5354\frac{53}{54}
5455\frac{54}{55}
5556\frac{55}{56}
n∈N\textcolor{blue}{n \in \mathbb{N}}
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