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?
