Long division of infinite series
Long division of numbers is usually taught in primary schools to children as one of the basic arithmetic skills. Consider, for example, \(486 \div 5\). It technically requires the child to atomize \(486\) into blocks of fives. There are a few ways to handle this problem. Since we know that \(90 \times 5 = 450\), one way to do \(486 \div 5\) is: $$\begin{align}486 &= (\mathbf{90} \times 5) + (480 - \mathbf{90} \times 5) + 6\\ &= (90 \times 5) + 30 + 6\\ &= (90 \times 5) + (6 \times 5) + (5 + 1)\\ &= (90 + 6 + 1)\times 5 + 1\\ &= 97 \times 5 + 1 \end{align}$$ We can also repeat the process with a different seed number, for instance, instead of \(90\), we can start with \(80\): $$\begin{align}486 &= (\mathbf{80} \times 5) + (480 - \mathbf{80} \times 5) + 6\\ &= (80 \times 5) + 80 + 6\\ &= (80 \times 5) + (50 + 30) + (5 + 1)\\ &= (80 + 10 + 6 + 1)\times 5 + 1\\ &= 97 \times 5 + 1 \end{align}$$ This essentially means that \(486\) can be a...
