Re: Idle Thoughts

Rudy Velthuis [TeamB] skrev:
"Henrick Hellström [StreamSec]" wrote: 
No. It is the limit of a sequence of finite series. Each one of the
finite series in the sequence approach 1/3 even closer. The limit
OTOH gets there.



The limit is 1/3. The sequence never reaches it. 0.3(bar) is the
(infinitely continuing) sequence, of which the limit is 1/3.

No. You might be confusing the concept of a sequence with the concept of a series. The /sequence/ is the numbers 0.9, 0.99, 0.999,...etc in order. Each number in this sequence can be expressed as a /series/ on the form num(n) = (sum of 9/(10 to the power of k) where k = 1 to n). The expression 0.9(bar) is short hand for a number with an infinite decimal expansion. This number is *not* part of the sequence of which 1 is the limit. The sequence contains numbers with a finite number of decimals only. Instead, the expression 0.999999... is, in turn, short hand for the limit of the previously mentioned sequence of finite series.
.

Relevant Pages

  • Re: Idle Thoughts
    ... The expression 0.9is short hand for a number with an infinite decimal expansion. ... This number is *not* part of the sequence of which 1 is the limit. ... is, in turn, short hand for the limit of the previously mentioned sequence of finite series. ...
    (borland.public.delphi.non-technical)
  • Re: Idle Thoughts
    ... It is an infinite series approaching 1/3 ever closer. ... (BTW none of the finite series in the sequence is infinite. ... mathematicians are reluctant to speak of "infinite series" since what they really talk about are limits of sequences of growing finite sequences. ...
    (borland.public.delphi.non-technical)
  • Re: Idle Thoughts
    ... It is the limit of a sequence of finite series. ... > finite series in the sequence approach 1/3 even closer. ... Prev by Date: ...
    (borland.public.delphi.non-technical)
  • Re: Review of Mueckenheims book.
    ... ....2222, they are actually infinitely distant elements of a sequence, ... smallest positive number, on the infinite scale. ... That's the ordering used by Cantor to prove their countability. ... If a set S is countable, then there is a total order < of S such that ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... I have always allowed that sequences can be countably infinite, ... like the sequence of standard naturals is. ... TO nor anyone else has produced any axiom system in which it can be ... at several removes from pure mathematics, ...
    (sci.math)
Loading