Leap years are determined by a scheme in which every $4$th year is a leap year, but every $4\cdot 25$th year is exempted, but every $4\cdot 25 \cdot 4$th year is reinstated $\ldots $ and there we stop, because that's good enough in practice to approximate the actual length of a year in terms of days. But what if we wanted to approximate arbitrarily well? In other words, can any real number $r$, $0<r<1$, be written in the form $$ r=\sum_{n=0}^{\infty} \frac{(-1)^n }{\prod_{i=0}^{n} a_i } $$ with $a_i $ a sequence of positive integers? For the earth at its current rate of rotation, where a fraction $r=0.242375$ of a day has to be represented, the existing sequence $a_0 =4$, $a_1 =25$, $a_2 =4$ merely has to be supplemented by $a_3 =20$ to reproduce this $r$.

Leap years are determined by a scheme in which every $4$th year is a leap year, but every $4\cdot 25$th year is exempted, but every $4\cdot 25 \cdot 4$th year is reinstated $\ldots $ and there we stop, because that's good enough in practice to approximate the actual length of a year in terms of days. But what if we wanted to approximate arbitrarily well? In other words, can any real number $r$, $0<r<1$, be written in the form $$ r=\sum_{n=0}^{\infty} \frac{(-1)^n }{\prod_{i=0}^{n} a_i } $$ with $a_i $ a sequence of positive integers? For the earth at its current rate of rotation, where a fraction $r=0.242375$ of a day has to be represented, the existing sequence $a_0 =4$, $a_1 =25$, $a_2 =4$ merely has to be supplemented by $a_3 =20$ to reproduce this $r$ to the full known precision.

Leap years are determined by a scheme in which every $4$th year is a leap year, but every $4\cdot 25$th year is exempted, but every $4\cdot 25 \cdot 4$th year is reinstated $\ldots $ and there we stop, because that's good enough in practice to approximate the actual length of a year in terms of days. But what if we wanted to approximate arbitrarily well? In other words, can any real number $r$, $0<r<1$, be written in the form $$ r=\sum_{n=0}^{\infty} \frac{(-1)^n }{\prod_{i=0}^{n} a_i } $$ with $a_i $ a sequence of positive integers? For the earth at its current rate of rotation, $a_0 =4$, $a_1 =25$, $a_2 =4$, $\ldots $

Leap years are determined by a scheme in which every $4$th year is a leap year, but every $4\cdot 25$th year is exempted, but every $4\cdot 25 \cdot 4$th year is reinstated $\ldots $ and there we stop, because that's good enough in practice to approximate the actual length of a year in terms of days. But what if we wanted to approximate arbitrarily well? In other words, can any real number $r$, $0<r<1$, be written in the form $$ r=\sum_{n=0}^{\infty} \frac{(-1)^n }{\prod_{i=0}^{n} a_i } $$ with $a_i $ a sequence of positive integers? For the earth at its current rate of rotation, where a fraction $r=0.242375$ of a day has to be represented, the existing sequence $a_0 =4$, $a_1 =25$, $a_2 =4$ merely has to be supplemented by $a_3 =20$ to reproduce this $r$.