 This is, of course, not news to mathematicians. The infinite sum produced by the Pasadena game is known as the alternating harmonic series, which is a conditionally convergent series. (A series \(a_n\) is conditionally convergent if \(\sum_{j=1}^{\infty} a_n\) converges but \(\sum_{j=1}^{\infty} \lvert a_n\rvert\) diverges.) Because of a theorem known as the Riemann rearrangement theorem, we know that if an infinite series is conditionally convergent, then its terms can always be rearranged such the sum converges to any finite number, or to \(+\infty\) or to \(-\infty\).