convergence of an infinite product (how to prove ?)

Question

Fix $C > 0 $ a constant and fix $n \in N$. Consider $\alpha \in (0,1)$ fixed.

I am reading a paper and the authors says:

For an arbitrary $r <1$ , the infinite product $$P = \prod_{j=0}^{\infty} (1+C 2^{n/2}(2^{-j}r)^{\alpha / 2})$$ always converges. How can I prove this? I dont know how to start. Someone can give me a help?

Thanks in advance!

Answer 1

The standard result for infinite products is that, provided no $p_n=-1$, $$ \prod_{n=0}^{\infty} (1+p_n) $$ converges if and only if $$ \sum_{n=0}^{\infty} |p_n| $$ does. In this case, you have $$ p_n = C 2^{n/2} (2^{-j}r)^{\alpha/2}, $$ on which you can use the ratio test to check the sum converges.