Sequence of Real Numbers

In Salas and Hille, page 642, a sequence of real valued numbers is defined to be a real-valued function whose domain consists of the postive integers. Usually, if W is the name of a function and x is in the domain of the function, W(x) is the name for the value of the function at x. By tradition, if W is a sequence, function values for sequences are written as W _n instead of W(n). For a sequence W, W(n) is not wrong---it's just not commonly written that way.

In other textbooks and in my own mathematical practice, sequences of real numbers are defined more generally: they might be real-valued functions defined on only part of the positive integers, and their domains might include 0 and some negative integers. Usually, the context will make clear the exact usage.

Here are some examples:
• a _n=1/n
• b _n=(-1)ⁿ
• c _n =cos(n ² )
• d _n =(1+1/n)ⁿ
• Sequences can be defined recursively. For example, one can start with e₁=1, say, and declare that in general e _n+1 =(e_n+5/(e_n ))/2 for each positive integer n. This specific example computes a sequence of rational numbers (ratios of integers) that converges to the square root of 5.
• Sequences can be defined by multi-term recursion. For example, one could start with f₁=1, f₂=1, and declare that f_n+2=f_n+ f_n+1 for all positive integers n. Thus, f₃=2, f₄=3, f₅=5, f₆=8, etc. For this example, there is also an explicit formula like the first 4 examples above.
• Sequences can be defined randomly. For example, for each positive integer n, you could flip a coin to decide that the value of the sequence is 0 or 1 (say, 0 if you get a head and 1 if you get a tail).

Your comments and questions are welcome. Please email ramsey@math.hawaii.edu.