our good calculus 2022年亚洲世界杯预选赛 know that $sum_{n=1}^infty frac 1n$diverges and that $sum_{n=1}^infty frac{(-1)^n}{n}$ converges. ourvery good 2022年亚洲世界杯预选赛 can even explain why $sum_{n=1}^infty frac{(-1)^{lfloor n /3 floor}}{n}$ converges. our stellar calculus2022年亚洲世界杯预选赛 may even be able to explain why $sum_{n=1}^infty frac{(-1)^{lfloor log n floor}}{n}$ diverges. in joint work withbruce reznick and monika serbinowska, we show that $$sum_{n=1}^infty frac{(-1)^{lfloor n sqrt{2} floor}}{n}$$converges. our proofs rely on diophantine properties of $sqrt{2}$, and donot apply (for example) if $sqrt{2}$ is replaced by$frac{sqrt{5}+1}{2}$.