solving a function with series

Question

Is it possible to find a solution for the following equation with respect to the parameter $\gamma_k$ where the equation is $$\alpha\gamma_k\beta_k-\alpha\ln\sum_{m=1}^K\exp(\gamma_m\beta_m)-\theta\gamma_k=0$$ where $\alpha$, $\theta$ and $\beta_i,i=\{1,...,K\}$ are constant?

Answer 1

Let $s_K = \sum_{m=1}^K \exp(\gamma_m \beta_m)$, then you have $$ \tag{*} \alpha \gamma_K \beta_K - \alpha \ln (\exp(\gamma_K \beta_K) + s_{K-1}) - \theta \gamma_K = 0 $$ If $\alpha = 0$, the problem is trivial, so assume $\alpha \ne 0$. Divide (*) by $\alpha$ and substitute $x = \gamma_K \beta_K$ and $t = \frac{\theta}{\alpha \beta_K}$ to get $$ \tag{**} x - \ln (e^x + s_{K-1}) - tx = 0 $$ and we would like to solve this for $x$.

Now you can expand $f(x) = \ln(e^x+s)$ into Maclaurin series (WolframAlpha) to get $$ \begin{split} \ln(e^x+s) \approx \ln(s + 1) &+ \frac{x}{s + 1} + \frac{s x^2}{2 (s + 1)^2} \\ &+ \frac{(s - 1) s x^3}{6 (s + 1)^3} + \frac{s (s^2 - 4 s + 1) x^4}{24 (s + 1)^4}\\ &+ \frac{s (s^3 - 11 s^2 + 11 s - 1) x^5}{120 (s + 1)^5} \\ &+ \frac{s (s^4 - 26 s^3 + 66 s^2 - 26 s + 1) x^6}{720 (s + 1)^6} + O(x^7) \end{split} $$ Take the quartic expansion and plug it into (**) and solve.