TITLE: How to calculate product $\prod_{k=0}^{n-1}\left (1+\frac{1}{2^{2^k}}\right)$?
QUESTION [8 upvotes]: How can I calculate the following product of series? $$ \prod_{k=0}^{n-1}\left (1+\frac{1}{2^{2^k}}\right)    $$....Can I take a geometric series and compare it with that?

REPLY [6 votes]: This is really the same as JimmyK4542's answer, but written in the general case.
If you expand the product, you find that it is the sum of all possible terms of the form
$$\frac{1}{2^{a_0 + a_1 2 + a_2 2^2 + \dots + a_{n-1}2^{n-1}}}$$
where each $a_i$ is either $0$ or $1$. Viewing the exponents as integers written in binary, they run though all integers $j = 0, 1, 2, \dots 2^n - 1$. We therefore find that the product is equal to
$$\sum_{j = 0}^{2^n - 1} \frac{1}{2^j} = 2 - \frac{1}{2^{2^n - 1}}.$$