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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)
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