TITLE: Evaluating $\prod_{n=1}^{\infty}\left(1+\frac{1}{2^n}\right)$
QUESTION [6 upvotes]: Compute:$$\prod_{n=1}^{\infty}\left(1+\frac{1}{2^n}\right)$$

I and my friend came across this product. Is the product till infinity equal to $1$?
If no, what is the answer?

REPLY [3 votes]: $$\prod_{n\geq 1}\left(1+\frac{1}{2^n}\right)=\exp\sum_{n\geq 1}\sum_{m\geq 1}\frac{(-1)^{m+1}}{m 2^{mn}}=\exp\sum_{m\geq 1}\frac{(-1)^{m+1}}{m(2^m-1)} $$
where the last series is (rapidly) convergent by Leibniz' test. It follows that the original product is between $\exp\left(\frac{121}{140}\right)$ and $\exp\left(\frac{3779}{4340}\right)$. By truncating the shown series at $m=30$ we get
$$\prod_{n\geq 1}\left(1+\frac{1}{2^n}\right)\approx 2.384231029.$$