A sequence is and ordered list of numbers, usually following some pattern like the “find the pattern” questions on IQ tests. We will study the properties of these sequences. For example, we can check whether the sequence converges to some limit.

Understanding sequences is also a prerequisite for understanding series, which is an important topic we will discuss in the next section.

• $\mathbb{N}$: The set of natural numbers $\{0, 1, 2, 3, \ldots \}$.
• $\mathbb{N}^*=\mathbb{N} \setminus \{0\}$:

The set of strictly positive natural numbers $\{1, 2, 3, \ldots \}$,

  which is the same as the above, but we skip zero.
* $a_n$: sequence of numbers $a_0, a_1, a_2, a_3, a_4, \ldots$.
  You can also think about each sequence as a function
  \[
     a: \mathbb{N} \to \mathbb{R},
  \]
  where the input is $n$ an integer (the //index// into the sequence) and
  the output is some number $a_n \in \mathbb{R}$.

Consider the following common sequences.

Consider a sequence in which successive terms differ by one: \[ 1, \ 2,\ 3, \ 4, \ 5, \ 6, \ \ldots \] which is described by the formula: \[ a_n = n, \qquad n \in \mathbb{N}^*. \]

More generally, an arithmetic sequence can start at any value $a_0$ and make jumps of size $d$ at each step: \[ a_n = a_0 + nd, \qquad n \in \mathbb{N}. \]

If we choose to make the sequence elements inversely proportional to the index $n$ we obtain the harmonic sequence: \[ 1, \ \frac{1}{2},\ \frac{1}{3}, \ \frac{1}{4}, \ \frac{1}{5}, \ \frac{1}{6}, \ \ldots \] \[ a_n = \frac{1}{n}, \qquad n \in \mathbb{N}^*. \]

More generally, we can define a $p$-sequence in which the index $n$ appears in the denominator raised to the power $p$: \[ a_n = \frac{1}{n^p}, \qquad n \in \mathbb{N}^*. \]

For example, when $p=2$ we get the sequence of inverse squares of the integers: \[ 1, \ \frac{1}{4}, \ \frac{1}{9}, \ \frac{1}{16}, \ \frac{1}{25}, \ \frac{1}{36}, \ \ldots. \]

If we use the index as an exponent to a fixed number $r$ we obtain the geometric series: \[ a_n = r^n, \ \ n \in \mathbb{N}, \] which is a sequence of the form \[ 1, r, r^2, r^3, r^4, r^5, r^6, \ldots. \]

Suppose we choose $r=\frac{1}{2}$, then the geometric series with this ratio will be: \[ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, \frac{1}{128}, \ldots. \]

\[ a_0 =1, a_1 = 1, \qquad \ a_n = a_{n-1} + a_{n-2}, \ \ n > 1. \] \[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \ldots. \]

We say a sequence $a_n$ converges to a limit $L$, or written mathematically: \[ \lim_{n \to \infty} a_n \ = \ L, \] if for large $n$ the sequence values get arbitrarily close to the value $L$.

More precisely, the limit notation means that for any choice of precisions $\epsilon>0$, we can pick a number $N_\epsilon$ such that: \[ | a_n - L | < \epsilon, \qquad \forall n \geq N_\epsilon. \]

The notion of a limit of a sequence is the same as that of a limit of a function. The same way we learned how to calculate which number the function $f(x)$ tends to for large $x$, we can study which number the sequence $a_n$ tends to for large $n$. Indeed, sequences are functions that are defined only at integer values of $x$.

The numbers in the Fibonacci sequence grow indefinitely large ($\lim_{n \to \infty} a_n = \infty$), but the ratio of $\frac{a_n}{a_{n-1}}$ converges to a constant: \[ \lim_{n \to \infty}\frac{a_n}{a_{n-1}} = \phi = \frac{1+\sqrt{5}}{2} \approx 1.618033\ldots, \] which is known as the golden ratio.

If a sequence $a_n$ is like a function $f(x)$, then we should be able to do calculus on it. We already saw we can take limits of sequences, but can we also compute derivatives and integrals of sequences? Derivatives are a no-go, because they depend on the function $f(x)$ being continuous and sequences are only defined for integer values. We can take integrals of sequences, however, and this is the subject of the next section.