A mathematical joke

OK this is not funny, just an absurdity caused by incorrect thinking.

Take the number 1 and double it, you get 2. Double it again, that's 4. Double again, 8, again, 16 and so on...

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ...

I seem to remember a story in Chinese history where an Emperor was duped by an Old Man and a chessboard. For some payment the Emperor agreed to place 1 grain of rice on the first square, 2 on the second, 4 on the third, 8 on the fourth and so on. Then the 64th square alone needs 9,223,372,036,854,775,808 grains of rice. Add up all the rice and it's twice that plus 1 (or is it -1) anyways it's about 18,446,744,073,709,551,616 grains. An impressive harvest.

However, had I been the Emperor I would have said "No. It's not enough payment. Let us make the chessboard infinitely large. How much do I owe you now Old Man?".

Let's work it out. We don't know what the final sum of all the rice grains will be so lets call it P, for Payment.

So to find P we have to add up the grains on every square, so

P = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 etc, etc to infinity.

How the hell do you do that? Think this way...

Double everything, ie multiply P by 2 and all the numbers by 2, then:

2P = 2 + 4 + 8 + 16 + 32 + 64 + 128 etc, etc to infinity.

Notice that each number in 2P has a counterpart in P, except for one (assuming the sequences continue to infinity) and that solitary digit is the number 1. It can't be in 2 times anything or it would be two.

Let's work out what the payment is. Subtract 2P from P, what do you get? What else can you get but 1?

P - 2P = 1

Therefore P = -1

"Old Man, you owe me a grain of rice"

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So where does the math go wrong?