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? 
 A: 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
$$(1+x)(1+x^2)…(1+x^{2^n})=\frac{1-x^{2^{n+1}}}{1-x}$$
A: Observe that
$$
(1+x)(1+x^2)...(1+x^{2n})=\frac{(1-x^2)}{1-x}(1+x^2)...(1+x^{2^n})
=\frac{(1-x^4)}{1-x}(1+x^4)...(1+x^{2^n})=\cdots=\frac{1-x^{2^{n+1}}}{1-x}\to\frac{1}{1-x}
$$
as $x^{2^{n+1}}\to 0$, whenever $\lvert x\rvert<1$.
A: An infinite product of terms each approaching one is not neccesarly one as well. For instance, see the classic:
$$e^x = \lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n$$ 
As to the actual limit, others have beat me to it :)
A: Try looking at the structures obtained for low values of $n$.
For example,
$(1+x)(1+x^2)=1+x+x^2+x^3$
$(1+x)(1+x^2)(1+x^4)=1+x+x^2+x^3+x^4+x^5+x^6+x^7$.
The resulting expressions can be simplified and understood using the behavior of sums of geometric series.
A: While other answers are based on the product $\prod (1 + x^{2^{n}})$, OP seems to be interested in the product $\prod (1 + x^{n})$. We can do some simplification like the following
\begin{align}
f(x) &= \prod_{n = 1}^{\infty}(1 + x^{n})\notag\\
&= \prod_{n = 1}^{\infty}\frac{1 - x^{2n}}{1 - x^{n}}\notag\\
&= \prod_{n = 1}^{\infty}\frac{1}{1 - x^{2n - 1}}\notag\\
\end{align}
This function has been studied by generations of mathematicians and belongs to the family of "Theta functions" and modern notation is to use $q$ in the place of $x$. Ramanujan studied the function $$g(q) = 2^{-1/4}q^{-1/24}\prod_{n = 1}^{\infty}(1 - q^{2n - 1}) = \frac{1}{2^{1/4}q^{1/24}f(q)}$$ and made the fundamental discovery that if $r$ is a positive rational number then $g(e^{-\pi\sqrt{r}})$ is a real algebraic number. These numbers are called Ramanujan's class invariants and denoted by $g_{r}$.
Thus for certain values of $q$ of type $q = e^{-\pi\sqrt{r}}, r \in \mathbb{Q}^{+}$ the value of $f(q)$ can be evaluated in a closed form in terms of $e, \pi$ and certain algebraic numbers. As an example 
\begin{align}
f(e^{-\pi}) &= (1 + e^{-\pi})(1 + e^{-2\pi})(1 + e^{-3\pi})(1 + e^{-4\pi})\cdots\notag\\
&= \frac{1}{(1 - e^{-\pi})(1 - e^{-3\pi})(1 - e^{-5\pi})\cdots}\notag\\
&= 2^{-1/8}e^{\pi/24}\notag
\end{align}
