More on 0.999…=1
In my last post, I discussed a particular video which I found to be more than a bit misleading. The discussion centered around a simple, but extremely counterintuitive notion of mathematics: the fact that the number 0.999…, or zero-point-nine-repeating, is equal to 1.
Well, as I mentioned, the very counterintuitive nature of the result led at least one of my readers to question its validity. As such, I thought I would lay out one proof of this concept, in order to make it easier for those who do not accept the result to pinpoint exactly where they disagree. I’ll break my proof down into numbered steps, to ease in that venture.
(1) Definition of 0.999…
By the symbol, 0.999…, I mean to say an infinite decimal expansion in which all digits to the right of the decimal place are 9’s. Mathematically, we can express this as:
(2) Partial Sums
Those of you who remember your Calculus might immediately recognize this Summation as a textbook example of a convergent geometric series. However, for those who do not, let’s work through the steps to determine the limit of this expression.
Provided the series converges, we say that the value of the summation is equal to that series’ limit.
Similarly, if convergent, the limit of the series is equal to the limit of the partial sums of the series. In general, the nth partial sum of our series can be seen to be:
So, as long as the series converges, we can see that:
If the partial sums form a convergent sequence, then the whole series converges. A convergent sequence is one which has an existent, finite limit.
So, now, if exists and is finite, then it follows that exists and is finite. And, if that is the case, then it is evident that our summation series from (1) converges.
We say that the limit of some function, , exists and is finite if, as k is made arbitrarily large, then becomes arbitrarily close to some Real number, L.
So, our question now becomes, as k is made any arbitrarily large value, does become arbitrarily close to any single Real number? It’s fairly obvious that, the larger k we utilize, the closer gets to 0. We can, in fact, make as close to 0 as we want, simply by choosing a large enough value of k.
In general, it will always be true that:
…for any Real number, k. Additionally, for any Real number r such that , we can choose a value of k which would make it true that:
And for all k, it is true that:
Since 0 is a finite, Real number, it is clear that the limit exists.
(5) Evaluating the Summation Series
Now, we have everything we need to evaluate our initial summation.