I recently read a proof that 1+2+4+8+16+… adds to -1. Relax! I’m not arguing it - I understood the proof overall and get the concept.
My only question has to do with a step in the proof. Here’s how it went:
Let’s call 1+2+4+8+16+… T.
So 2T = 2+4+8+16+32 . . .
The author observed that 2T is T, only without the 1 at the front so therefore, 2T = T-1. Therefore, T = -1. Got it, I really do, assuming that when subtracting one infinite sum, from the other, one aligns them like this:
2T-T = (0-1) + (2-2) + (4-4) + … = -1
But is there a rule that that is the way you have to do the subtraction? If that’s the convention, no worries. but I don’t get why you can’t do the subtraction like this:
So, as I say, looking only for an explanation of the rules/conventions of subtracting infinite sums, not trying to open a can of worms like the infamous thread on the sum of natural numbers = -1/12!
I’m no mathematician, but I find that assertion highly suspect. If it is true for infinite sums, should it therefore not also be true for some subset of that infinite sum?
If you start out from the very beginning, where T=1, the difference of the resulting sums is never -1, but T. Which is what we’d expect from the operation 2T-T. At what point along the line does the result suddenly become -1?
In a sense, these are the same thing, since the series 2+4+8+… ought to be equivalent to the series 0+2+4+8+… And if you were working with convergent series, you would be able to subtract term-by-term like this. (Cite: see the blue box about halfway down the page.)
In the context of this book, the “proof” cited by the OP is intended to be analogous to the arguments some people give for why 0.999… = 1, showing that those arguments “lack something.”
Thudlow Boink pointed out upthread that the proof is actually cited as an example of thinking that is “lacking something”, so the point now is to find the something that’s lacking.
And I think that something is – I don’t know what the actual mathematical term is for it, but he’s applying an operation to one side of the equation without doing the same thing to the other side of the equation.
To begin, we have the infinite sum T. We first have to arrive at the second sum, which is 2T. We get that through multiplying T by 2.
2 * T = 2T
Now, as Ellenberg notes, the difference between 2T and T appears to be 1 (which it actually isn’t; it increases with each iteration of the sum as it approaches infinity). So if we simply subtract 1 from the left-hand side of the equation:
(2 * T) - 1 = 2T
But simplifying that out, we get
2T - 1 = 2T
which is mathematically impossible. In order to maintain consistency, we have to apply operations of equal magnitude to both sides of the equation (in this case, decreasing both infinite sums by 1).
Appearance masks form, making reality appear to be something it actually isn’t.
It seems to me that the underlying error is the assumption that the sum T is a number, so that values like T - 1 and 2T are thus also numbers that we can apply the usual rules of addition and subtraction to.
The problem here is one of understanding what the initial statement says, which I will restate here:
“The sum of unity and twice every integer from unity to and including infinity converges to negative unity”
Implicit in the initial statement is the concept of “infinity”, and that cannot be ignored. Further, it cannot be denied that “infinity” is not manipulable under the ordinary arithmetic operations and properties (associativity, distributivity, commutativity); one may not “multiply infinity by two and produce two infinities”.
The initial statement expresses a series which does not converge to a solution, because as ever larger partial sums are added to the “partial sum so far” the resulting partial sum continues to grow (and at an ever increasing rate!). Therefore, under the standard definition of an infinite sum, which is “the sum of all partial sums in the series”, there can be no convergence to any solution.
The initial statement is rejected.
Well, the something is that he’s performing operations that would be perfectly legitimate if done to finite sums, or even to convergent infinite series, on series that do not converge.
If you just had T = 1 + 2 + 4 + 8 + 16, it would be perfectly correct to say that
2T = 2(1 + 2 + 4 + 8 + 16) = 2 + 4 + 8 + 16 + 32.
It would also be perfectly correct to say that T–1 = (1 + 2 + 4 + 8 + 16) – 1
= 2 + 4 + 8 + 16.
But of course, that doesn’t match 2 + 4 + 8 + 16 + 32 (the 32 is missing).
You may want to read the wikipedia article Divergent series. Basically there are many different ways to sum a divergent series, and different methods may give different answers, as the OP has discovered. Roughly speaking, this means that divergent series don’t have a well-defined sum, which perhaps should not be surprising. I haven’t read the Ellenberg book, but I hope that he went on to explain the nuances of this issue, and not simply state that the sum of the series in the OP is flatly equal to -1, which is misleading at best.
So here’s what I learned (feel free to correct me if I didn’t get it right):
Saying an infinite divergent series “is” something depends on (as a noted amateur semiotician once phrased it) “what your definition of is is”
There are a metric tonne of methods of adding (and so, I take it by implication, subtracting) infinite divergent series
Thudlow has read every popular math book known to man
Divergent series are an invention of the devil (I saw that line in the Wikipedia article, and then when I reviewed Ellenberg, I saw it was in there, too)
So next question: I haven’t reviewed the epic threads on the sum of natural numbers being -1/12, (see here and herebut couldn’t the same point (there are a variety of ways of adding divergent series) apply to that as well? I don’t remember anybody making that point, or else they wouldn’t go on for several pages, like they do. So doesn’t the same point apply to that series as this one?
And while Thudlow is correct that the section of Ellenberg does invoke the idea that proofs like this, .9999…=1, and he also throws in 1+1-1+1-1+. . . = ½, “lack something”, he is talking about the mathematical proof’s ability to “address the anxious uncertainty induced by the claim”. And he is invoking all this stuff to introduce the idea that intuition and mathematical rigour can be at odds with one another.
Anyway, thank you all for fighting my ignorance on the matter!
Think of infinity as a number the same as division by zero (which, coincidentally, yields infinity). As others point out, standard math does not apply. Half infinity is infinity. twice infinity is infinity. Infinity minus infinity is undefined (the crux of the OP question), since twice infinity is infinity.
Infinities are not very large finites. They are a wholly different thing that often behave in confusing and non-intuitive ways (when your intuition is developed by working with finite things).
