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Limit of a Sequence

Q: Can Mathematica evaluate the limit of an infinite series where the domain is the natural  numbers? Limit assumes that the function is continuous. For example,
[Graphics:../Images/index_gr_55.gif]
[Graphics:../Images/index_gr_56.gif]
is true if [Graphics:../Images/index_gr_57.gif] but not true if [Graphics:../Images/index_gr_58.gif] is a positive integer (in which case the limit is 0).
A: Andrzej Kozlowski (andrzej@tuins.ac.jp) answers.
What you really want is to find the limit of a sequence of real numbers,  i.e., a function [Graphics:../Images/index_gr_59.gif] from the positive integers to the reals ([Graphics:../Images/index_gr_60.gif]). Using the fact that [Graphics:../Images/index_gr_61.gif] as [Graphics:../Images/index_gr_62.gif] runs over the integers is equal to [Graphics:../Images/index_gr_63.gif], where [Graphics:../Images/index_gr_64.gif] is an integer, you can get your answer as follows.
[Graphics:../Images/index_gr_65.gif]
[Graphics:../Images/index_gr_66.gif]
When using this method you should make sure that [Graphics:../Images/index_gr_67.gif] is defined in the range of values over which you are taking the sum.  Sometimes the form of the answer is complicated, even in cases which can be  easily solved by using Limit. For example, consider the limit of the function
[Graphics:../Images/index_gr_68.gif]
as [Graphics:../Images/index_gr_69.gif] runs over the integers starting with any [Graphics:../Images/index_gr_70.gif]. Using Sum gives a complicated answer.
[Graphics:../Images/index_gr_71.gif]
[Graphics:../Images/index_gr_72.gif]
Applying FullSimplify yields the correct answer,
[Graphics:../Images/index_gr_73.gif]
[Graphics:../Images/index_gr_74.gif]
which, in this case, is the same as that given by the continuous limit.
[Graphics:../Images/index_gr_75.gif]
[Graphics:../Images/index_gr_76.gif]


Converted by Mathematica      May 8, 2000

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