How to define a mathematical sequence

Description

In mathematics, a sequence is an infinite list of values, typically real numbers, often written $a_0,a_1,a_2,\dots$, 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

Using SymPy, in Python

View this solution alone.

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 $\mathrm{\infty }$).

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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

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

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

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seq[20]

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

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seq[:10]

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

You can compute the limit of a sequence,

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limit( a_n, n, oo )

${\displaystyle 0}$

Content last modified on 24 July 2023.

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Topics that include this task

• Bentley University GR526

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