Evaluate $\lim_{n\to\infty}(1+x)(1+x^2)\cdots(1+x^{2n}),|x|<1$ I get that the limit is
$$\lim\limits_{n\to\infty}f(x)=1$$
because
$$\lim\limits_{n\to\infty}(1+x^{2n})=1,\,\,\lvert x\rvert<1.$$
Is this right? 

Hint: Multuply by $1-x$. Then you get
$$(1-x)(1+x)(1+x^2)…(1+x^{2^n})=(1-x^2)(1+x^2)…(1+x^{2^n}) =\\
(1-x^4)(1+x^4)…(1+x^{2^n})=...=(1-x^{2^{n+1}})$$
Thus
$$(1+x)(1+x^2)…(1+x^{2^n})=\frac{1-x^{2^{n+1}}}{1-x}$$
