1.1 Generalities on sequences
ACTIVITY 1.1
Fold once an A4 paper, what is the fraction that represents the part you are seeing?
Fold it twice, what is the fraction that represents the part you are seeing?
What is the fraction that represents the part you are seeing if you fold it ten times?
What is the fraction that represents the part you are seeing if you fold it n times?
Write a list of the fractions obtained starting from the first until the nth fraction.
CONTENT SUMMARY
Let us consider the following list of numbers: \( 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{n}, \ldots \). The terms of this list are compared to the images of the function \( f(x) = \frac{1}{x} \). The list never ends, as the ellipsis indicates. The numbers in this ordered list are called the terms of the sequence. In dealing with sequences, we usually use subscripted
letters, such as \( u_1 \) to represent the first term, \( u_2 \) for the second term, \( u_3 \) for the third term, and so on such as in the sequence \( f(n) = u_n = \frac{1}{n} \).
However, in the sequence such as \( {u_n} : u_n = \sqrt{n-3} \), the first term is u3 as the previous are not possible, in the sequence \( {u_n} : u_n = 2n-5 \), the first term is \( u_0 \).
Definition
A sequence is a function whose domain is the set of natural numbers.
The terms of a sequence are the range elements of the function.
It is denoted by \( u_1, u_2, u_3, \ldots, u_{n-1}, u_n\) and shortly \( \{ u_n \} \). We can also write \( \{u_1, u_2, u_3, \ldots, u_{n-1}, u_n\} \). The dots are used to suggest that the sequence continues indefinitely, following the obvious pattern.
The numbers \( u_1, u_2, u_3, \ldots, u_{n-1}, u_n\) in a sequence are called terms of the sequence. The natural number \( n \) is called term number and value \( u_n \) is called a general term of a sequence and the term \( u_1 \) is the initial term or the first term.
As a sequence continues indefinitely, it can be denoted as \( \{ u_n \}^{\infty}_{n=1} \).
The number of terms of a sequence (possibly infinite) is called the length of the sequence.
Notice
- Sometimes, the term number, n, starts from 0. In this case terms of a sequence are \( u_0, u_1, u_2, \ldots, u_{n-1}, u_n, \ldots \) and this sequence is denoted by \( \{ u_n \}^{+\infty}_{n=0} \). In this case the initial term is \( u_0 \).
- A sequence can be finite, like the sequence \( 2, 4, 8, 16, \ldots, 256 \).
The empty sequence \( \{ \} \) is included in most notions of sequences, but may be excluded depending on the context.
Usually a numerical sequence is given by some formula \( u_n = f(n) \), permitting to find any term of the sequence by its number \( n \); this formula is called a general term formula.
A second way of defining a sequence is to assign a value to the first (or the first few) term(s) and specify the nth term by a formula or equation that involves one or more of the terms preceding it. Sequences defined this way are said to be defined recursively, and the rule or formula is called a recursive formula.