Silhouette wrote:I don't know why this wasn't an open/shut thread. A nice little trick to figure out, for sure, but it doesn't warrant in depth discussion unless you don't understand it. It certainly doesn't throw numbers into doubt.

Exactly.

Silhouette wrote:gib wrote:I'm interested in this:

1 + 2 + 3 + 4 + 5 + 6 + 7 + .... = -1/12

This one might warrant discussion though.

It happens when you mess around with infinite series, notably "s = 1-2+3-4+5-6+7-..." and what happens when you add it to itself in a certain way (i.e. add the 1st term to the 2nd, the 2nd to the 3rd and so on) and noting that it results in "t = 1-1+1-1+1-1+1-...", which you have to accept equals a 1/2, because it's an average of whether you "stop" at the last even number in the series or the last odd number in the series.

So adding s to itself... what would this look like?

(1-2+3-4+5-6+7...) + (1-2+3-4+5-6+7...) = 1 + (1-2) + (3-2) + (3-4) ... = 1 - 1 + 1 -1 ... = 1/2

^ Ah, look at that! ^

Silhouette wrote:Accepting these, you can say that 2s = t = 1/2, therefore s = 1/4. I see.

Then you can subtract "s" from the infinite series in question of "u = 1+2+3+4+5+6+7+...", the first term from the first term, the second from the second and so on - nothing special here. You get "u-s = 0+4+0+8+12+0+16+...", so you can factor out the 4 and get "u-s = 4(1+2+3+4+5+6+7+...)", which is "u-s=4u".

This simplifies to "-s = 3u", we know s = 1/4 so "-1/4 = 3u", making our infinite series "u = -1/12".

I see how you get there.

Silhouette wrote:Obviously it rides on the problem of "t". I don't see why you can't add "s" to itself in the way I described above... infinite series extend infinitely so it doesn't matter what order you add each number together. The rest is just standard rearrangement and substitution.

I follow.

I don't think the infinite series "t" has a valid answer, which makes possible such things as "1+2+3+4+5+6+7+..." = -1/12. I'm pretty sure you can make it equal other different values too.

I'm inclined to agree. Saying that 1 - 1 + 1 - 1 + 1... = 1/2 seems to make good folk sense, but I don't think it follows from any arithmetic rule. The series 1 - 1 + 1 - 1 + 1... can be rewritten as (1 + 1 + 1 ...) + (-1 - 1 - 1 - 1 ...), which means infinity plus negative infinity which would give you zero (and maybe you can't even say that). I think we have to assume that in order for the rules of arithmetic to hold, your expressions have to at least be complete. Saying 4 + 2 + 1 = 7, for example, says that when you have 4 of something, and you also have 2 of that something, and one more of that something, you have 7 of that something all together. But how are you supposed to say how much of something you have if you just never stop listing of how many in each group of somethings you have (well, you'd have infinity, but what happens when you include negatives)?