General problem
$$ \sum_{i=1}^n \frac{a_i + x}{b_i + x} = 0 $$
Is it possible for solve for $x$?
Some context
I've hit a road block in my derivation... At this point, I need to pull the model parameter $\theta$ out of the summation.
Is it possible to separate $\theta$ from the summation in the below equation (when $\theta$ appears in the denominator)?
$$ \sum_{i=1}^n \frac{(\nu + 1)(1 - p_i)(y_i + \theta (p_i - 1) - \eta p_i)}{\sigma^2 \nu + (y_i + \theta (p_i - 1) - \eta p_i)^2} $$
Any suggestions?
More context
I took the derivative with respect to $\theta$ of a monstrous log-probability, and I am hoping to solve for the optimal value of $\theta$.
After setting the derivative to zero, I get the following monstrosity... which I hope I can solve provided that I can pull $\theta$ out of the summation.
$$ 0 = \frac{\kappa (1 - \omega) Exp(\theta; \kappa) + (\theta - \mu) \tau^{-2} \omega Norm(\theta; \mu, \tau^2)} {(1 - \omega) Exp(\theta; \kappa) - \omega Norm(\theta; \mu, \tau^2)} - \sum_{i=1}^n \frac{(\nu + 1)(1 - p_i)(y_i + \theta (p_i - 1) - \eta p_i)}{\sigma^2 \nu + (y_i + \theta (p_i - 1) - \eta p_i)^2} $$
Is it hopeless to pull $\theta$ out of the denominator in a summation of fractions, and I should try something else?