1 + 10 + 100 + 1000 + = -1/9

[Unit]

$$S = 1 + 10 + 100 + 1000 + 10000 + ...$$

$$10S = 10 + 100 + 1000 + 10000 + 100000 + ...$$

$$S - 10S = (1 + 10 + 100 + 1000 + 10000 + ...) - (10 + 100 + 1000 + 10000 + ...)$$

$$-9S = 1 + (10 - 10) + (100 - 100) + (1000 - 1000) + (10000 - 10000) ...$$

$$-9S = 1 + 0 + 0 + 0 + 0 + 0 ...$$

$$-9S = 1$$

$$S = -1/9$$


What's wrong (or right) with this?

Thanks,
Unit

 

[CRGreathouse]

http://en.wikipedia.org/wiki/Ramanujan_summation

 

[Unit]

Thanks!
Ii also just found this: http://motls.blogspot.com/2007/09/zeta-function-regularization.html

 

[Hurkyl]

Also, I believe that sum converges as an ordinary infinite sum in the 2-adics and the 5-adics.

(And, of course, it does not converge as an ordinary infinite sum in the reals!)

 

[CRGreathouse]

I would have [intuitively] expected it to converge in all the p-adics. Am I wrong?

 

[Hurkyl]

In any other p-adic field, the terms don't converge to zero!

 

[CRGreathouse]

In any other p-adic field, the terms don't converge to zero!

 

[Char. Limit]

I'm pretty sure that you can't pair up terms in an infinite sum.

 

[Redbelly98]

$$S = 1 + 10 + 100 + 1000 + 10000 + ...$$
.
.
.
What's wrong (or right) with this?

Thanks,
Unit

I'm pretty sure that you can't pair up terms in an infinite sum.

The way I remember it, proofs like this actually say something like:

If S exists, then S = 1 + 10 + 100 + ...​

So if S does not exist, then the remaining statements do not necessarily hold true.

 

[Unit]

If S exists, then S = 1 + 10 + 100 + ...​

So if S does not exist, then the remaining statements do not necessarily hold true.

Brilliant! I had completely forgotten about variables and their related hypothetical syllogisms. Thanks!