**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 e
_{1}=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
f
_{1}=1, f_{2}=1, and declare that f_{n+2}=f_{n}+ f_{n+1}for all positive integers n. Thus, f_{3}=2, f_{4}=3, f_{5}=5, f_{6}=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).
- a
Your comments and questions are welcome. Please email ramsey@math.hawaii.edu. |