# 1 + 10 + 100 + ... = -1/9

July 27, 2011 6:45 AM Subscribe

Mathfilter: In Gelfand's exceptional Algebra text, he is talking about the formula 1 + x + x^2 + x^3 + ... = 1/(1 - x). He uses the Achilles racing the tortoise model when first introducing the formula. He discusses the case x = 10, whence 1 + 10 + 100 + 1000 + ... = -1/9.

Then he says the following: "Is it possible to give a reasonable interpretation of the (absurd) statement 'Achilles will meet the turtle after running -1/9 meters?' (Hint: Yes, it is.) "

I have puzzled over this for awhile and am not sure what he had in mind. Does the hive mind have any ideas?

Then he says the following: "Is it possible to give a reasonable interpretation of the (absurd) statement 'Achilles will meet the turtle after running -1/9 meters?' (Hint: Yes, it is.) "

I have puzzled over this for awhile and am not sure what he had in mind. Does the hive mind have any ideas?

Achilles runs -.001, turtle runs -.01 -- turtle at -11/100, Achilles at -111/1000

Achilles runs -.01, turtle runs -.1 -- turtle at -1/10, Achilles at -11/100

Achilles runs -.1, turtle runs -1 -- turtle at 0, Achilles at -1/10

START: Achilles at 0, turtle at 1

Achilles runs 1, turtle runs 10 -- Achilles at 1, turtle at 10

Achilles runs 10, turtle runs 100 --Achilles at 11, turtle at 110.

Achilles can't catch the turtle going at positive speeds. But at negative speeds, both Achilles and the turtle are converging on -1/9ths, just like they converge on 10/9ths if Achilles is 10x faster than the turtle.

posted by michaelh at 7:10 AM on July 27, 2011

Achilles runs -.01, turtle runs -.1 -- turtle at -1/10, Achilles at -11/100

Achilles runs -.1, turtle runs -1 -- turtle at 0, Achilles at -1/10

START: Achilles at 0, turtle at 1

Achilles runs 1, turtle runs 10 -- Achilles at 1, turtle at 10

Achilles runs 10, turtle runs 100 --Achilles at 11, turtle at 110.

Achilles can't catch the turtle going at positive speeds. But at negative speeds, both Achilles and the turtle are converging on -1/9ths, just like they converge on 10/9ths if Achilles is 10x faster than the turtle.

posted by michaelh at 7:10 AM on July 27, 2011

LogicalDash: There's various cases where you can do mildly misleading things with such sequences. As discussed in this previous AskMe for example, the sum of all integers is sometimes said to be equal to -1/12.

Plenty of caveats apply, of course, but it's a disturbingly useful result in some bits of physics.

posted by edd at 7:19 AM on July 27, 2011

Plenty of caveats apply, of course, but it's a disturbingly useful result in some bits of physics.

posted by edd at 7:19 AM on July 27, 2011

The infinite series only converges for -1 < x < 1.

posted by Obscure Reference at 7:36 AM on July 27, 2011

posted by Obscure Reference at 7:36 AM on July 27, 2011

**michaelh**is the closest, I think. The original race can be rephrased as follows:

Achilles and a tortoise are both running at constant velocity to the right. At a particular moment in time, Achilles is at the location x = 0 along the track, and the tortoise is at the location x = 1 (to the right along the track.) Achilles runs at a speed of 1, and the tortoise runs at a speed of 1/10. At what location do they meet?The answer can be found, either by algebra (write down the equations for the motion of both Achilles and the tortoise, and set them equal to each other) or by the infinite-series method, to be the spot x = 10/9.

Now for the speedy-tortoise version:

Achilles and a tortoise are both running at constant velocity to the right. At a particular moment in time, Achilles is at the location x = 0 along the track, and the tortoise is at the location x = 1 (to the right along the track.) Achilles runs at a speed of 1, and the tortoise runs at a speed of 10. At what location do they meet?If you run through the algebraic method, your equations give you the solution -1/9. This seems a little weird until you realize that I didn't say that the race started at the "particular moment" above. In other words, if you had come in a little earlier, you would have seen the tortoise pass Achilles at the location -1/9, and then (a moment later) the tortoise at x = 1 and Achilles at x = 0. In other words, if you "ran the clock backwards" with the given speeds, you would find that Achilles and the tortoise met up

*before*Achilles got to the spot x = 0. This is, at least, a "reasonable interpretation" of the sum 1 + 10 + 100 + ... = -1/9, though as noted above there are issues with the formal mathematical definitions of "convergence" in this context. This kind of quick-and-dirty interpretation is the kind of thing that physicists tend to do cavalierly, even if (because?) it gives the mathematicians the screaming abdabs.

posted by Johnny Assay at 7:43 AM on July 27, 2011 [5 favorites]

That was fun, thanks michael and Johnny. You can bioassay my titratrions any time you please.

posted by Buckt at 1:47 PM on July 27, 2011

posted by Buckt at 1:47 PM on July 27, 2011

My thanks to everyone also. Gelfand is fond of physical applications, so I suspect the marked answers are correct.

posted by wittgenstein at 6:05 AM on July 28, 2011

posted by wittgenstein at 6:05 AM on July 28, 2011

This thread is closed to new comments.

posted by LogicalDash at 6:52 AM on July 27, 2011