>>16722360 (OP)[math]
\delta \, \epsilon \left ( 0,1 \right ) \\
\displaystyle
\prod_{k=0}^{n} \left ( 1 + \delta ^{2^{k}} \right )
= (1+ \delta)(1+ \delta ^{2})(1+ \delta ^{4}) \cdots (1+ \delta ^{2^{n-1}})(1+ \delta ^{2^{n}}) \\
(1- \delta) \prod_{k=0}^{n} \left ( 1+ \delta ^{2^{k}} \right )
= (1- \delta)(1+ \delta)(1+ \delta ^{2})(1+ \delta ^{4}) \cdots (1+ \delta ^{2^{n-1}})(1+ \delta ^{2^{n}}) \\
= (1- \delta ^{2})(1+ \delta ^{2})(1+ \delta ^{4}) \cdots (1+ \delta ^{2^{n-1}})(1+ \delta ^{2^{n}}) \\
= (1- \delta ^{4})(1+ \delta ^{4}) \cdots (1+ \delta ^{2^{n-1}})(1+ \delta ^{2^{n}})
= (1- \delta ^{2^{n}})(1+ \delta ^{2^{n}}) = 1- \delta ^{2^{n+1}} \\
\\
\boxed{(\delta ^{2^n})^2 = \delta ^{2 \cdot 2^n}=\delta^{2^{n+1}}}
\\
\displaystyle
\prod_{k=0}^{n} \left ( 1+ \delta ^{2^{k}} \right )
= \dfrac{1- \delta^{2^{n+1}}}{1- \delta} \\
\displaystyle
\lim_{n \to \infty} \prod_{k=0}^{n} \left (1+ \delta ^{2^{k}} \right )
= \lim_{n \to \infty} \dfrac{1- \delta^{2^{n+1}}}{1- \delta} = \dfrac{1}{1- \delta}
[/math]