### SICP: Broken Math

Here's a great little math trick from SICP:

v = 1 + 2 + 4 + 8 + ...Let's play around with that equation a bit:

v - 1 = 2 + 4 + 8 + ...But that's the same as the first line. Hence, we see that:

(v - 1) / 2 = 1 + 2 + 4 + 8 + ...

(v - 1) / 2 = vSolving for v, we get:

v = -1Crazy, eh? Sussman said:

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

Arguments that are based on convergence are flaky, if you don't know the convergence beforehand. You can make wrong arguments. You can make deductions as if you know the answer and not be stopped somewhere by some obvious contradiction.I think the bad assumption in this case is thinking that some v could exist and be a normal integer. Clearly, there is no such normal integer v that equals 1 + 2 + 4 + 8 + ...

## Comments

Sussman is right, we have to be very careful when handling limits.

But for this one, it seems rather natural that v will not be an integer.

Sn = 1 + 2 + 3 + ... + n

Gauss started by writing the numbers like this :

Sn = 1 2 3 4 5 ... n

then in reversed order :

Sn = n n-1 n-2 2 1

What's the point ? He could now add each column and see that they all give n+1:

2*Sn = n+1 n+1 n+1 ... n+1

As with have n equal terms, 2*Sn is easily computed :

2*Sn = = n*(n+1)

So, Sn = n*(n+1)/2

Nice no ?!

Haha, I thought of that too ;)

> Why do you call it a trick?

I didn't know what else to call it ;)

Ah, yes, I know that story. The funny is when I showed my wife this "trick", she came up with the same story ;) So let me see:

n = infinity

(infinity (infinity - 1)) / 2 = infinity

(But note, it's the same "kind" of infinity, since there's a 1-1 mapping between the two.)

where they give a bibliography, including a book by G. H. Hardy.

I think this particular sum is due to Euler.