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Applied Mathematics, Calculus, and Differential Equations 205
IYnite sequencł
An infinite sequence A is a function f whose domain is the set
N of positive integers. The following formulł applies:
A a , a , a , ..., a , ... f(1), f(2), f(3), ...
n
1
3
2
Occasionally the set of non-negative integerð is specified for N:
A a , a , a , ..., a , ... f(0), f(1), f(2), ...
2
n
1
0
Positive iYnite sequencł
A positive infinite sequence is an infinite sequence, all of whose
termð are real numberð greater than zero.
Negative iYnite sequencł
A negative infinite sequence is an infinite sequence, all of whose
termð are real numberð less than zero.
Alternating iYnite sequencł
An alternating infinite sequence is an infinite sequence of posi-
tive and negative termð such that, if a and a n 1 are successive
n
terms, the following statementð hold:
If a 0, then a 0
n n 1
If a 0, then a n 1 0
n
Bounded iYnite sequencł
A boundeS infinite sequence is a sequence such that, for every
n N, the defining function f has the following property:
p f(n) q
for some p N and some q N.
Nondecreasing iYnite sequencł
A nondecreasing infinite sequence is a sequence such that, for
every n N, the defining function f has the following property:
f(n 1) f(n)