TITLE: Infinite product limit
QUESTION [7 upvotes]: Suppose we have the infinite product $2(1/2)(2^4)(1/2^8)(2^{16})\dots$ I have a hunch that the infinite product is $0$ despite partial product being strictly positive. Am I correct? If so, then how?
Thank you ahead of time!

REPLY [5 votes]: Note that we have: $2 \cdot \frac 12 \cdot 2^4 \cdot \frac{1}{2^8} \cdot... = 2 \cdot 2^{-1} \cdot {2^4} \cdot 2^{-8} \cdot ... = 2^{1-1+4-8+16-...}$
As the sum of the exponents doesn't converge we have that the product doesn't converge too.