How to define a mathematical sequence (in Python, using SymPy)

Task

In mathematics, a sequence is an infinite list of values, typically real numbers, often written $a_{0}, a_{1}, a_{2}, \dots$ , or collectively as $a_{n}$ .

(Let’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:

How to define a mathematical series

Solution

This answer assumes you have imported SymPy as follows.

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 $\infty$ ).

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

$[1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots]$

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

seq[20]

$\frac{1}{21}$

seq[:10]

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

You can compute the limit of a sequence,

lim_{n \to \infty} a_{n} .

limit( a_n, n, oo )

$0$

Content last modified on 24 July 2023.

See a problem? Tell us or edit the source.

Contributed by Nathan Carter (ncarter@bentley.edu)