I have the following infinite sum, with $0<\theta<1$ and $0<\theta_3\leq 1$ and $\theta_3\geq (2\theta-1)/\theta^2$, $$1-\sum_{x=0}^{+\infty}\theta^{2x+1}\cdot\theta_3^{x(x+1)}\cdot(1-\theta_3^{x+1}\cdot\theta)$$ which is actually a probability.
Is it possibile to prove that this sum is an increasing function of $\theta_3$ for any possible choice of $\theta$? I tried by computing the first order derivative, but I was not able to prove that the latter is always greater than zero...