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#233 Trace in Float caused Fatal Error ←	↑	→ #235 Where is RadicalExtension and RationalRadical?

#234 limit(-1/2)^n

Edit detail for #234 limit(-1/2)^n revision 1 of 2

	1 2
Editor: page Time: 2011/04/07 07:59:53 GMT-7
Note: check limits

changed:
-
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:
\begin{axiom}
)version
limit( 2 + (-2/%pi)^n, n=%plusInfinity )
\end{axiom}

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?

From unknown Sun Nov 13 11:27:38 -0600 2005
From: unknown
Date: Sun, 13 Nov 2005 11:27:38 -0600
Subject: An answer to my question, from William Sit.
Message-ID: <20051113112738-0600@wiki.axiom-developer.org>

William Sit <wyscc@cunyvm.cuny.edu> 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:
\begin{axiom}
limit( (-1)^n,  n=%plusInfinity )
limit( (n+ 1)/n,  n=%plusInfinity )
limit( (-1)^n * (n + 1)/n,  n=%plusInfinity )
\end{axiom}

From kratt6 Thu Dec 20 02:54:46 -0800 2007
From: kratt6
Date: Thu, 20 Dec 2007 02:54:46 -0800
Subject: 
Message-ID: <20071220025446-0800@axiom-wiki.newsynthesis.org>

Category: Axiom Mathematics => Axiom Library

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:

axiom

)version

Value = "FriCAS 2010-12-08 compiled at Thursday March 10, 2011 at 19:56:28 "
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:

axiom

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

$\label{eq2}\mbox{\tt "failed"}$

(2)

Type: Union("failed",...)

axiom

limit( (n+ 1)/n,  n=%plusInfinity )

$\label{eq3}1$

(3)

Type: Union(OrderedCompletion?(Fraction(Polynomial(Integer))),...)

axiom

limit( (-1)^n * (n + 1)/n,  n=%plusInfinity )

$\label{eq4}\mbox{\tt "failed"}$

(4)

Type: Union("failed",...)

... --kratt6, Thu, 20 Dec 2007 02:54:46 -0800 reply

Category: Axiom Mathematics => Axiom Library