18. Sequences

a1. Definition

A sequence is simply an ordered list of numbers, and the numbers in the list are called the terms of the sequence. A sequence is finite or infinite if the number of terms is finite or infinite. If we don't say whether it is a finite or infinite sequence, we normally mean an infinite sequence.

Each of the following lists is a sequence:

1.  \(1,2,3,4,5,\cdots\)	The counting numbers.	An infinite sequence.
2.  \(10\),  \(9\),  \(8\),  \(7\),  \(6\),  \(5\),  \(4\),  \(3\),  \(2\),  \(1\),  \(0\)	Countdown sequence... Blast Off!	A finite sequence.
3.  \(3,1,4,1,5,9,2,6,5,3,\cdots\)	The digits of \(\pi\).	An infinite sequence.
4.  \(1,\dfrac{1}{2}\,\dfrac{1}{4}\,\dfrac{1}{8}\,\dfrac{1}{16}\,\dfrac{1}{32}\,\cdots\)	Keep halving.	An infinite sequence.

In general, we denote the terms of a sequence as a letter with an integer subscript:
  \(a_1,a_2,a_3,a_4,\cdots\)
and we frequently give a formula for the \(n^\text{th}\) term.

Write out the first \(5\) terms of the sequence \(b_n=\dfrac{1}{2n+1}\) for \(n=1,2,3,\cdots\)

\[ b_1=\dfrac{1}{3},\;b_2=\dfrac{1}{5},\;b_3=\dfrac{1}{7},\;b_4=\dfrac{1}{9}, \;b_5=\dfrac{1}{11},\;\cdots \]

Normally the terms are counted starting from \(n=1\), as above, but they can start from any number, as in this exercise:

To indicate an entire sequence, we can enclose the general term in braces: \(\{ a_n \}\) If necessary, we can also indicate the range of the index: \(\{ a_n\}_{n=1}^\infty\) For example, the sequence in the preceding exercise would be denoted \(\left\{\dfrac{n}{n-1}\right\}_{n=2}^\infty\).