Evaluating limit of exponential product

Question

I have the following product to evaluate as a limit: $$\lim\limits_{n\rightarrow\infty}\prod_{k=-1}^n(r^{kr^k})$$ where $0<r<1$. Upon graphing this for various values of $r$, it seems that there is a finite limit depending on the value of $r$, namely decreasing $r$ gives a greater limit and vice versa. Any advice on how to evaluate this?

Answer 1

Taking logarithm of the product you get $\log r\sum_{k=-1}^nkr^k$. Now $\sum_{k=0}^nkr^{k-1}$ is the sum$\frac{d}{dx}\sum_{k=1}^nx^k$ evaluated at $x=r$. Now for $|x|<1$, $\sum_{k=1}^nx^k=\frac{1-x^{n+1}}{1-x}$.

Can you do the rest?