telescoping products

the “catalyst” tool
Utilizing difference-of-squares in telescoping products that are geometric series.

Example. Simplify the product

    \[(1+\fr{a})(1+\fr{a^2})(1+\fr{a^4})\dots(1+\fr{a^{2^{100}}}).\]

Call this P, and multiply it by the catalyst \ \ 1-\fr{a}.

    \begin{align*} (1-\fr{a})P &=(1-\fr{a})(1+\fr{a})(1+\fr{a^2})\dots(1+\fr{a^{2^{100}}}) \\ &=(1-\fr{a^2})(1+\fr{a^2})(1+\fr{a^4})\dots(1+\fr{a^{2^{100}}}) \\ &=(1-\fr{a^4})(1+\fr{a^4})(1+\fr{a^8})\dots(1+\fr{a^{2^{100}}}) \\ &\vdots \\ &=(1-\fr{a^{2^{100}}})(1+\fr{a^{2^{100}}}) \\ &=1-\fr{a^{2^{101}}}. \\ \text{Hence } \ \ P&=\frac{1-(1/a^{2^{101}})}{1-(1/a)} \end{align*}