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Submitted by : (unknown) at: 2007-11-17T22:13:08-08:00 (15 years ago)
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Another problem I have is that taking the limit of an expression containing (-1)^n always returns "failed", where my TI-89 Titanium calculator will give a finite limit. For example:

limit( 2 + (-2/%pi)^n, n=%plusInfinity ) ===> "failed"

... but the TI-89t returns 2.

This version of Axiom (FriCAS) gives:

Value = "FriCAS 1.2.1 compiled at Monday July 1, 2013 at 23:26:54 " limit( 2 + (-2/%pi)^n, n=%plusInfinity )

\label{eq1}\mbox{\tt "failed"}(1)
Type: Union("failed",...)

The TI-89t says that the limit of (-1)^n as n approaches infinity is -1, implying that it believes that infinity is an odd number. That kind of makes sense to me, since if you divide infinity in half, you still have infinity, and you keep adding 1 to get to infinity, making it odd. If infinity is even then the answer should be 1, and if we can't know if infinity is even or odd, then the answer is uncertain or undefined.

On the other hand, the TI-89t says that lim ( (-1)^n * (n + 1)/n ) is undefined. But it already told me that lim (-1)^n = -1, and that lim (n + 1)/n = 1. If the limit of a product is the product of the limits of the factors, then lim ( (-1)^n * (n + 1)/n ) should be -1, right?

So, who's right?

An answer to my question, from William Sit. --unknown, Sun, 13 Nov 2005 11:27:38 -0600 reply
William Sit responded via emai, and wrote, in part:

Mathematically, the limit is 2 since (-2/%pi) has absolute value less than 1, and hence (-2/%pi)^n converges to 0. So TI-89t is correct and Axiom is wrong.

In the second case, TI-89t is wrong to say that limit (-1)^n is -1. The limit does not exist because the sequence (-1)^n oscillates between 1 and -1. There is no number L (the assumed limit) such that given any epsilon > 0, there is a natural number N such that |(-1)^n - L| < epsilon for all n > N.

At least Axiom gets this right:

limit( (-1)^n,  n=%plusInfinity )

\label{eq2}\mbox{\tt "failed"}(2)
Type: Union("failed",...)
limit( (n+ 1)/n,  n=%plusInfinity )

Type: Union(OrderedCompletion?(Fraction(Polynomial(Integer))),...)
limit( (-1)^n * (n + 1)/n,  n=%plusInfinity )

\label{eq4}\mbox{\tt "failed"}(4)
Type: Union("failed",...)

Category: Axiom Mathematics => Axiom Library

Name: #234 limit((-1/2)^n,n=%plusInfinity) => #234 limit(-1/2)^n

  Subject:   Be Bold !!
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