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 =(en+5/(en ))/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 fn+2=fn+ fn+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).

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