Find limits of series: $\lim_{n \to \infty} (1+x)(1+x^2)\cdots(1+x^{2n})$

Question

Find $$\ \lim_{n \to \infty} (1+x)(1+x^2)...(1+x^{2n}), \ |x| <1$$ What i have done is $$\ \lim_{n \to \infty}(1+x)(1+x^2)...(1+x^{2n})\ =\ \lim_{n \to \infty}\ (1+x)\ \ \ \lim_{n \to \infty}\ (1+x^2)...\ \lim_{n \to \infty}(1+x^{2n})\ \ $$

Is this correct?

Answer 1

I'm assuming you mean $|x|<1$, and that the product you are trying to evaluate is given by $$ a_n=(1+x)(1+x^2)(1+x^4)(1+x^8)\cdots (1+x^{2^n}). $$ Then consider $(1-x)a_n$: $$ \begin{align*} (1-x)a_n&=(1-x^2)(1+x^2)(1+x^4)\cdots(1+x^{2^n})\\ & = (1-x^4)(1+x^4)\cdots(1+x^{2^n})\\ &=\cdots =1-x^{2^{n+1}}, \end{align*} $$ so therefore $$ \lim_{n\to\infty}a_n=\lim_{n\to\infty}\frac{1-x^{2^{n+1}}}{1-x}=\frac{1}{1-x}. $$