# How to define a mathematical sequence

## Description

In mathematics, a sequence is an infinite list of values, typically real numbers, often written $a_0,a_1,a_2,\ldots$, or collectively as $a_n$.

(Letâ€™s assume that sequences are indexed starting with index 0, at $a_0$, even though some definitions start with index 1, at $a_1$, instead.)

How can we express sequences in mathematical software?

Related tasks:

## Using SymPy, in Python

This answer assumes you have imported SymPy as follows.

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from sympy import * # load all math functions
init_printing( use_latex='mathjax' ) # use pretty math output

Sequences are typically written in terms of an independent variable $n$, so we will tell SymPy to use $n$ as a symbol, then define our sequence in terms of $n$.

We define a term of an example sequence as $a_n=\frac{1}{n+1}$, then
build a sequence from that term. The code `(n,0,oo)`

means that $n$
starts counting at $n=0$ and goes on forever (with `oo`

being the SymPy
notation for $\infty$).

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var( 'n' ) # use n as a symbol
a_n = 1 / ( n + 1 ) # formula for a term
seq = sequence( a_n, (n,0,oo) ) # build the sequence
seq

$\displaystyle \left[1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots\right]$

You can ask for specific terms in the sequence, or many terms in a row, as follows.

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seq[20]

$\displaystyle \frac{1}{21}$

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seq[:10]

$\displaystyle \left[ 1, \ \frac{1}{2}, \ \frac{1}{3}, \ \frac{1}{4}, \ \frac{1}{5}, \ \frac{1}{6}, \ \frac{1}{7}, \ \frac{1}{8}, \ \frac{1}{9}, \ \frac{1}{10}\right]$

You can compute the limit of a sequence,

\[\lim_{n\to\infty} a_n.\]1

limit( a_n, n, oo )

$\displaystyle 0$

Content last modified on 24 July 2023.

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