Compute $\lim_{n\to\infty}(1+a)(1+a^2)(1+a^4)...(1+a^{2^n})$? Compute
$$\lim_{n\to\infty}(1+a)(1+a^2)(1+a^4)...(1+a^{2^n})$$
Where $|a|<1$.

$$(1-a)f=(1-a)(1+a)(1+a^2)(1+a^4)(1+a^8)...(1+a^{2^n})=(1-a^{2^{n+1}})$$
Use $(x-y)(x+y)=x^2-y^2$ sucsessively to get
$$f_n=\frac{1-a^{2^{n
+1}}}{1-a}.$$
As $|a|<1$
$$\lim_{n \rightarrow \infty} f_n= \frac{1}{1-a}.$$
