Page 32 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 32

Sequences













                     DEFINITION OF A SEQUENCE
                        A sequence is a set of numbers u 1 ; u 2 ; u 3 ; ... in a definite order of arrangement (i.e., a correspondence
                     with the natural numbers) and formed according to a definite rule.  Each number in the sequence is
                     called a term; u n is called the nth term. The sequence is called finite or infinite according as there are or
                     are not a finite number of terms.  The sequence u 1 ; u 2 ; u 3 ; ... is also designated briefly by fu n g.

                     EXAMPLES.  1. The set of numbers 2; 7; 12; 17; ... ; 32 is a finite sequence; the nth term is given by
                                   u n ¼ 2 þ 5ðn   1Þ¼ 5n   3, n ¼ 1; 2; ... ; 7.
                                2. The set of numbers 1; 1=3; 1=5; 1=7; .. . is an infinite sequence with nth term u n ¼ 1=ð2n   1Þ,
                                   n ¼ 1; 2; 3; ... .
                        Unless otherwise specified, we shall consider infinite sequences only.


                     LIMIT OF A SEQUENCE
                        A number l is called the limit of an infinite sequence u 1 ; u 2 ; u 3 ; ... if for any positive number   we can
                     find a positive number N depending on   such that ju n   lj <  for all integers n > N.In such case we
                     write lim u n ¼ l.
                          n!1
                     EXAMPLE . If u n ¼ 3 þ 1=n ¼ð3n þ 1Þ=n,the sequence is 4; 7=2; 10=3; ... and we can show that lim u n ¼ 3.
                                                                                             n!1
                        If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent.A
                     sequence can converge to only one limit, i.e., if a limit exists, it is unique.  See Problem 2.8.
                        A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence
                     u 1 ; u 2 ; u 3 ; ... has a limit l if the successive terms get ‘‘closer and closer’’ to l.  This is often used to
                     provide a ‘‘guess’’ as to the value of the limit, after which the definition is applied to see if the guess is
                     really correct.


                     THEOREMS ON LIMITS OF SEQUENCES
                        If lim a n ¼ A and lim b n ¼ B, then
                          n!1           n!1
                        1.  lim ða n þ b n Þ¼ lim a n þ lim b n ¼ A þ B
                            n!1         n!1     n!1
                        2.  lim ða n   b n Þ¼ lim a n   lim b n ¼ A   B
                            n!1         n!1     n!1
                        3.  lim ða n   b n Þ¼ ð lim a n Þð lim b n Þ¼ AB
                            n!1         n!1    n!1
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