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MathGroup Archive 1999

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Re: Limit of an Integer Function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg18799] Re: [mg18767] Limit of an Integer Function
  • From: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>
  • Date: Thu, 22 Jul 1999 08:19:21 -0400
  • Sender: owner-wri-mathgroup at wolfram.com

What you really want is for Mathematica to find the limit of a sequence of
real numbers, i.e. a function f from the integers to the reals. Actually,
this ability is built into Mathematica but in a not very explicit way. What
you can do is  use the fact that Lim[f[n],n->Infinity] as n runs over the
integers is equal to  Sum[(f[n+1]-f[n],{n,a,Infinity}]+f[a], where a is any
integer. Thus you can get your answer as follows:

In[2]:=
Sum[Sin[(n + 1)*Pi] - Sin[n*Pi], {n, 1, Infinity}] + Sin[Pi]
Out[2]=
0

When using this method you should make sure that f[n] is always defined in
the range of values over which you are taking the sum. Also Mathematica may
sometimes produce a complicated answer even in cases which can be easily
solved by using Limit. For example, consider the limit of the function

In[1]:=
f[n_] := (3n^2 + 1)/(n^2 - 1)

as n runs over the integers starting with any n>1. In this case the answer
is the same as given by the continuous limit

In[2]:=
Limit[f[n], n -> Infinity]
Out[2]=
3

Using Sum gives a complicated answer:
In[3]:=
Sum[Evaluate[Simplify[f[n + 1] - f[n]]], {n, 2, Infinity}] + f[2]
Out[3]=
13
-- - 4 RootSum[(1 + #1) (2 + #1) (3 + #1) (4 + #1) & ,
3

    -((PolyGamma[0, -#1] (1 + 2 (2 + #1))) /

        ((1 + #1) (2 + #1) (3 + #1) + (1 + #1) (2 + #1) (4 + #1) +

          (1 + #1) (3 + #1) (4 + #1) + (2 + #1) (3 + #1) (4 + #1))) & ]

Applying FullSimplify shows that this is indeed correct:

In[3]:=
FullSimplify[%]
Out[3]=
3

--
Andrzej Kozlowski
Toyama International University
JAPAN
http://sigma.tuins.ac.jp
http://eri2.tuins.ac.jp


----------
>From: Phil Mendelsohn <mend0070 at tc.umn.edu>
To: mathgroup at smc.vnet.net
>To: mathgroup at smc.vnet.net
>Subject: [mg18799] [mg18767] Limit of an Integer Function
>Date: Tue, Jul 20, 1999, 6:33 AM
>

> I seem to have found a blind spot in my knowledge of how to get
> Mathematica to evaluate limits.
>
> I want to evaluate the limit of a function where the domain is a member
> of the Natural numbers, such as infinite series.  It seems that Limit
> assumes that the function is continuous.
>
> For example, if I asked
>
> Limit[Sin[ n Pi ],n-> Infinity], mathematica would return:
> Interval[{-1,1}].  This is true if n is a member of the Reals, but not
> true if n is a positive integer (in which case the limit would be 0.]
>
> Is there another function I should use?  Or would it be nice to specify
> the domain of the function as a feature request?
>
>
> Phil Mendelsohn


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