Product of Sums: Show that the following is a Polynomial by converting it into standard form. $$\prod_{k=0}^n (1+x^{2^k})$$
The given expression simplifies to $(1+x)(1 + x^2)...(1 + x^{2^n})$
I am not able to proceed further. How do I express this in Summation form?

We have
$$\prod_{k = 0}^n (1 + x^{2^k}) = \prod_{k = 0}^n \frac{1 - (x^{2^k})^2}{1 - x^{2^k}} = \prod_{k = 0}^n \frac{1 - x^{2^{k+1}}}{1 - x^{2^k}} = \frac{1 - x^{2^{n+1}}}{1 - x},$$
and 
$$\frac{1 - x^{2^{n+1}}}{1 - x} = 1 + x + x^2 + \cdots + x^{2^{n+1} - 1}.$$
Thus
$$\prod_{k = 0}^n (1 + x^{2^k}) = 1 + x + x^2 + \cdots + x^{2^{n+1} - 1}.$$
