The king's reversal in the classical chess problem?

Question

I was talking to my math teacher about the classical problem where the king loses a chess match against a peasent and he is asked to give him $2^n$ grains for the $n$'th square (where the squares are numbered $1$ through $64$).This seems to be a classical example of the power of exponentiation.

However my teacher tells me there is a reversal of this problem where the king tells him that he will go further and pretend as if the board was infinite. The peasent agrees being very greedy, but apparently this is bad for the peasent? I am confused here, did I understand incorrectly, why is this good for the King?

Answer 1

If the field-to-field factor is $x$ (instead of $2$), the total number of grains for $n$ fields is $\frac{x^{n+1}-1}{x-1}$. If $|x|<1$ and we let $n$ grow to infinity, this value approaches $\frac1{1-x}$. So if we abuse(!) this limit result and apply it also for the case at hand, i.e. with $x=2$, then there should be $\frac1{1-2}=-1$ grains on an infinite chess board - the peasent owes the king one grain.

Another reason to justify the result: If we assume all grains placed and then double the content of each field, and then prepend the infinite board with a single field with one grain on it, we obtain the original situation. Hence the number of grains must be a solution of $2\cdot N+1=N$. The only finite solution is indeed $N=-1$.

Answer 2

Consider the Binary Kingdom, where by royal decree, binary computers are the ultimate source of truth for mathematical disputes. Every calculation must be performed using $n$-bit registers for some $n$. If two subjects disagree on a calculation because they used a different number of bits, then the subject who used more bits wins in court.

In this kingdom, $1+2+4+\cdots+2^{63}=2^{64}-1$ is a large amount of grain indeed. Someone operating with fewer than $64$ or fewer bits cannot tell the difference between this number and $-1$, but anyone with at least $65$ bits is aware of its enormity.

On the other hand, the infinite sum $1+2+4+\cdots$ equals $-1$ using any number of bits. So the court exchequer may claim that the peasant owes the king one grain, and the peasant cannot prove otherwise!

This kingdom has effectively mandated that $2$-adic numbers are the default setting for arithmetic. The court, by declaring that the subject with the most bits wins, is performing an inverse limit over their calculations. In the $2$-adic norm, we indeed have $|2|_2=\frac12<1$, so the geometric series converges. Unfortunately, there is no good ordering of the entire set of $2$-adic numbers, so economic jurisprudence in this kingdom is going to become pretty thorny when infinite transactions are allowed. Who can be held responsible for settling a debt if the amount is neither positive nor negative?

In modern civil society, we mostly prefer the real numbers, which have a nice ordering. Of course, in the real numbers, the infinite series $1+2+4+\cdots$ diverges. An aggressive resummation method would be needed to produce the value $-1$, such as analytic continuation of the function defined by the power series $1+2x+4x^2+\cdots$ to $x=1$.