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Topic: Warning to visitors to sci.math
Replies: 266   Last Post: Mar 19, 2012 9:13 AM

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Rotwang

Posts: 1,679
From: Swansea
Registered: 7/26/06
Re: Warning to visitors to sci.math
Posted: Mar 14, 2012 8:30 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 14/03/2012 09:11, DavidW wrote:
> [...]
>
> I thought this was of interest in this matter. I quote from "Calculus" by
> Michael Spivak (Addison-Wesley, World Student Series, 1967), Chapter 22
> ("Infinite series"), p. 388:
> - - - - -
> Infinite sequences were introduced in the previous chapter with the specific
> intention of considering their "sums"
> a1 + a2 + a3 + . . .
> in this chapter. This is not an entirely straightforward matter, for the sum of
> infinitely many numbers is as yet completely undefined. What can be defined are
> the "partial sums"
> Sn = a1 + . . . + an
> and the infinite sum must presumably be defined in terms of these partial sums.
> ... If there is to be any hope of computing the infinite sum a1 + a2 + a3 + . .
> . the partial sums Sn should represent closer and closer aproximations as n is
> chosen larger and larger. This last assertion amounts to little more than a
> sloppy assertion of limits: the "infinite sum" a1 + a2 + a3 + ...ought to be
> lim(n -> oo) Sn.
> - - - - -
>
> So he admits that infinite sums are sloppy,


No he doesn't. He says that his last assertion ("the partial sums Sn
should represent closer and closer aproximations as n is chosen larger
and larger") amounts to little more than a sloppy assertion of limits,
which it does. There's nothing sloppy about the /actual definition/ of
an infinite sum, namely that a sequence (a_n | n in N) has infinite sum
s iff the sequence of partial sums (sum_{i = 0}^n a_i | n in N) has
limit s, or equivalently, iff for every real number epsilon > 0 there
exists a natural number m such that |s - sum_{i = 0}^n a_i| < epsilon
for every natural number n > m.


> and the terms he puts in quotes
> indicate that they really aren't accurate terms because of the inherent
> incompleteness of infinite sequences. Contrasting with examples of sequences
> that are not summable he goes on to give an "acceptable definition" for a
> summable sequence. All this vagueness doesn't look good for infinity.


A summable sequence is defined to be a sequence that has an infinite
sum, as I defined above. What do you find vague about that definition?


> Yet, as I
> said earlier, I am willing to accept sequences that converge as the number of
> terms increases because, if that is the best definition of a number, that's
> just how that number is. e is such a number.


What do you mean by e? I know what I mean when I use that letter, but
for the reason I've already given it's apparent that you must mean
something different. What is it?


Date Subject Author
3/7/12
Read Warning to visitors to sci.math
DavidW
3/7/12
Read Re: Warning to visitors to sci.math
adamk
3/8/12
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Brian Chandler
3/8/12
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Richard Tobin
3/8/12
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Shmuel (Seymour J.) Metz
3/8/12
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amzoti
3/9/12
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micky
3/8/12
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Frederick Williams
3/8/12
Read Re: Warning to visitors to sci.math
DavidW
3/8/12
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DavidW
3/8/12
Read Re: Warning to visitors to sci.math
DavidW
3/8/12
Read Re: Warning to visitors to sci.math
J. Antonio Perez M.
3/8/12
Read Re: Warning to visitors to sci.math
DavidW
3/8/12
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J. Antonio Perez M.
3/9/12
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DavidW
3/9/12
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Brian Chandler
3/9/12
Read Re: Warning to visitors to sci.math
DavidW
3/9/12
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DavidW
3/9/12
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Brian Chandler
3/9/12
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DavidW
3/9/12
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Frederick Williams
3/9/12
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DavidW
3/9/12
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Frederick Williams
3/9/12
Read Re: Warning to visitors to sci.math
DavidW
3/9/12
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J. Antonio Perez M.
3/10/12
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Frederick Williams
3/10/12
Read Re: Warning to visitors to sci.math
DavidW
3/10/12
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DavidW
3/10/12
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Jerry Friedman
3/10/12
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Don Phillipson
3/10/12
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Harrison Hill
3/12/12
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Tom P
3/12/12
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DavidW
3/12/12
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Prai Jei
3/12/12
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J. Antonio Perez M.
3/12/12
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Peter Moylan
3/12/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
3/13/12
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Rotwang
3/13/12
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DavidW
3/13/12
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Rotwang
3/14/12
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DavidW
3/14/12
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Rotwang
3/14/12
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J. Antonio Perez M.
3/14/12
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DavidW
3/14/12
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Horand.Gassmann@googlemail.com
3/14/12
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J. Antonio Perez M.
3/14/12
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DavidW
3/14/12
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3/14/12
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DavidW
3/14/12
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J. Antonio Perez M.
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DavidW
3/14/12
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Rotwang
3/14/12
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Rotwang
3/15/12
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Jussi Piitulainen
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DavidW
3/14/12
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Rotwang
3/14/12
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DavidW
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DavidW
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3/14/12
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DavidW
3/14/12
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Rotwang
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DavidW
3/15/12
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Rotwang
3/15/12
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DavidW
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DavidW
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DavidW
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Rotwang
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Rotwang
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Rotwang
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DavidW
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3/14/12
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Transfer Principle
3/15/12
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FredJeffries@gmail.com
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Transfer Principle
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Daryl McCullough
3/16/12
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J. Antonio Perez M.
3/14/12
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ross.finlayson@gmail.com
3/12/12
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Stan Brown
3/13/12
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R H Draney
3/13/12
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Daryl McCullough
3/13/12
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Snidely
3/13/12
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Lanarcam
3/13/12
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3/13/12
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Richard Tobin
3/13/12
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Peter Moylan
3/14/12
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R H Draney
3/14/12
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Transfer Principle
3/14/12
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Rotwang
3/15/12
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Transfer Principle
3/15/12
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Rotwang
3/14/12
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ross.finlayson@gmail.com
3/14/12
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J. Antonio Perez M.
3/15/12
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Transfer Principle
3/16/12
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Daryl McCullough
3/17/12
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Michael Stemper
3/17/12
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Peter Moylan
3/14/12
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Michael Stemper
3/14/12
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Tim Little
3/13/12
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3/13/12
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3/13/12
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3/13/12
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3/14/12
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3/14/12
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3/13/12
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3/14/12
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3/14/12
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3/15/12
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3/15/12
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3/16/12
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quasi
3/16/12
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3/16/12
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quasi
3/15/12
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3/10/12
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3/12/12
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3/12/12
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3/13/12
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3/13/12
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3/13/12
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3/14/12
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3/10/12
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3/11/12
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DavidW
3/12/12
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3/9/12
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3/8/12
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3/8/12
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3/8/12
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3/10/12
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3/10/12
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3/10/12
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3/10/12
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3/10/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/11/12
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3/12/12
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3/12/12
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DavidW
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3/12/12
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3/12/12
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3/12/12
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3/12/12
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3/12/12
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DavidW
3/12/12
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3/12/12
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quasi
3/12/12
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DavidW
3/12/12
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3/12/12
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3/12/12
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DavidW
3/12/12
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Brian Chandler
3/12/12
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DavidW
3/12/12
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DavidW
3/12/12
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DavidW
3/12/12
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J. Antonio Perez M.
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DavidW
3/13/12
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Michael Stemper
3/13/12
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DavidW
3/13/12
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3/13/12
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J. Antonio Perez M.
3/13/12
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DavidW
3/13/12
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Richard Tobin
3/12/12
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3/12/12
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3/12/12
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3/12/12
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3/12/12
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3/12/12
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3/13/12
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3/15/12
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3/9/12
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3/9/12
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3/9/12
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3/10/12
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3/10/12
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3/13/12
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3/14/12
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3/14/12
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3/15/12
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3/15/12
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3/19/12
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3/9/12
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