Thursday, May 5, 2016

How can 1+2+3+4+... = -1/12?

There's this assertion that has become somewhat famous, as many YouTube videos have been made about it, that the infinite sum of all positive integers equals -1/12. Most people just can't accept it because it seems completely nonsensical and counter-intuitive.

One has to understand, however, that this is not just a trick, or a quip, or some random thing that someone came up at a whim. In fact, historical world-famous mathematicians came to that same conclusion independently of each other, using quite different methodologies. For example, some of the most famous mathematicians of all history, including Leonhard Euler, Bernhard Riemann and Srinivasa Ramanujan, all came to that same result, independently, and using different methods. They didn't just assign the value -1/12 to the sum arbitrarily at a whim, but they had solid mathematical reasons to arrive to that precise value and not something else.

And it is not the only such infinite sum with a counter-intuitive result. There are infinitely many of them. There is an entire field of mathematics dedicated to studying such divergent series. A simple example would be the sum of all the powers of 2:

1 + 2 + 4 + 8 + 16 + 32 + ... = -1

Most people would immediately protest to that assertion. Adding two positive values gives a positive value. How can adding infinitely many positive values not only not give infinity, but a negative value? That's completely impossible!

The problem is that we tend to instinctively think of infinite sums only in terms of its partial finite sums, and the limit that these partial sums approach when more and more terms are added to it. However, this is not necessarily the correct approach. The above sum is not a limit statement, nor is it some kind of finite sum. It's a sum with an infinite number of terms, and partial sums and limits do not apply to it. It's a completely different beast altogether.

Consider the much less controversial statement:

1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ... = 2

ie. the sum of the reciprocals of the powers of 2. Most people would agree that the above sum is valid. But why?

To understand why I'm asking the question, notice that the above sum is not a limit statement. In other words, it's not:

This limit is a rather different statement. It is saying that as more and more terms are added to the (finite) sum, the result approaches 2. Note that it never reaches 2, only that it approaches it more and more, as more terms are added.

If it never reaches 2, how can we say that the infinite sum 1 + 1/2 + 1/4 + ... is equal to 2? Not that it just approaches 2, but that it's mathematically equal to it? Philosophical objections to that statement could ostensibly be made. (How can you sum an infinite amount of terms? That's impossible. You would never get to the end of it, because there is no end. The terms just go on and on forever; you would never be done. It's just not possible to sum an infinite number of terms.)

Ultimately, the notation 1 + 1/2 + 1/4 + ... = 2 is a convention. A consensus that mathematics has agreed upon. In other words, we accept the notion that a sum can have an infinite number of terms (regardless of some philosophical objections that could be presented against that idea), and that such an infinite sum can be mathematically equal to a given finite value.

While in the case of convergent series the result is the same as the equivalent limit statement, we cannot use the limit method with divergent series. As much as people seem to accept "infinity" as some kind of valid result, technically speaking it's a nonsensical result, when we are talking about the natural (or even the real) numbers. It's meaningless.

It could well be that divergent sums simply don't have a value, and this may have been what was agreed upon. Just like 0/0 has no value, and no sensible value can be assigned to it, likewise a divergent sum doesn't have a value.

However, it turns out that's not the case. When using certain summation methods, sensible finite values can be assigned to divergent infinite sums. And these methods are not arbitrarily decided on a whim, but they have a strong mathematical reasoning for them. And, moreover, different independent summation methods reach the same result.

We have to understand that a sum with an infinite number of terms just doesn't behave intuitively. It does not necessarily behave like its finite partial sums. The archetypal example often given is:

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... = pi/4

Every term in the sum is a rational number. The sum of two rational numbers gives a rational number. No matter how many rational numbers you add, you always get a rational number. Yet this infinite sum does not give a rational number, but an irrational one. The infinite sum does not behave like its partial sums, nor does it follow the same rules. In other words:

"The sum of two rational numbers gives a rational number": Not necessarily true for infinite sums.

"The sum of two positive numbers gives a positive number": Not necessarily true for infinite sums.

Even knowing all this, you may still have hard time accepting that an infinite divergent sum of positive values not only gives a finite result, but a negative one. We are so strongly attached to the notion of dealing with infinite sums in terms of its finite partial sums that it's hard for us to put aside that approach completely. It's hard to accept that infinite sums do not behave the same as finite sums, nor can they be approached using the same methods.

In the end, it's a question of which mathematical methods you accept on a philosophical level. Just consider that these divergent infinite sums and their finite results are serious methods used by serious professional mathematicians, not just some trickery or wordplay.

More information about this can be found eg. at Wikipedia.

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