Leibniz (if I recall what I read yesterday afternoon) was once provided the sum S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ....

and calculated since 2S = 2 + 1 + 1/2 + 1/4 + 1/8 + ... then 2 S = 2 + S therefore S = 2. (While Aristotle concluded it could be no number other than 2.)

But the procedure does not work for all geometric series as if Z = 1 + 2 + 4 + 8 + 16 + ... and 2Z = 2 + 4 + 8 + 16 + 32 + ... = Z - 1 leads to the absurd conclusion that 2 Z = Z - 1 and therefore Z = -1.

Thus Leibniz's mistake was in assuming the infinite geometric sum existed as a real number before doing operations on it. However, since the infinite geometric sum is a number if the absolute value of the ratio between successive terms is less than 1, when this condition is met (as it is in repeating decimal fractions) then Leibniz's trick is justified.

Since we should agree that 0.999... is real number and 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + ... by definition, I submit we have not fallen into the same trap as Leibniz's guess before formalization of the ideas of limits and supremum.

Chinglu's last objection seems to be that no infinite sum can exist at all because doing the additions one-at-a-time would be a never-ending process. I agree, but do not accept that one-at-a-time is the only way to do addition. Lots of operations in set theory work even for infinite sets. Since all the terms being added are all positive, the limit of the partial sums as the number of terms grows without limit is the same number as the supremum of the set of all partial sums; the infinite sum (if it exists as a real number) can be no other number than this value. Further the digits of a decimal fraction themselves as bounded below by 0 and bounded above by 9. Thus every decimal fraction of the form $$a_0.a_1a_2a_3...$$ is a number bounded below by $$a_0$$ and bounded above by $$a_0 + 1$$.

Chinglu's quote-mining of his source in [post=3142405]post #1625[/post] is misleading, in that the value of the infinite series is actually introduced on that page.

So when you have an infinite series (a sum over all the terms of an infinite sequence), the value of that sum is defined to be the limit of the partial sums if such a sum exists.An infinite series is an expression like this:

S = 1 + 1/2 + 1/4 + 1/8 + ...

...

It is clear what the pattern is: the n-th partial sum is

$$S_n = 2 - 1/2^n$$

When n gets larger and larger, Sn gets closer and closer to the number 2. When a sequence $$S_n$$ gets closer and closer and closer to a given number S, we say that S is the limit of the Sn's and we write

lim( $$S_n$$ ) = S.

For geometric series that limit exists if the ratio has an absolute value less than 1. And for the series formed from a sequence formed by the pair-wise multiplication of a bounded sequence and a sequence that would lead to a converging series already, that new series is also going to converge. 0 to 9 are the bounds on a decimal sequence of digits and 1/10 is clearly the geometric ratio involved, so all decimal fractions of infinite length are in fact infinite sums that form real numbers.