Can $\frac{\sum_{k = 0}^n e^{\beta_1 \left( t_k + 2 t_n \right)}}{\sum_{k = 0}^n e^{\beta_1 \left( t_k + t_n \right)}}$ be simplified?

Question

Can the expression $$\dfrac{\sum_{k = 0}^n e^{\large\beta_1 \left( t_k + 2 t_n \right)}} {\sum_{k = 0}^n e^{\large\beta_1 \left( t_k + t_n \right)}}\;\;\text{ where}\;t_i\ge0\;\text { and}\;\;\beta_1\ge0$$ be simplified to avoid overflow due to limited precision arithmetic? The sequence $t_k$ is strictly increasing, that is, $t_{k+1}>t_k$.

Answer 1

I would find the value of $k=k_m$ such that $\beta_1 (t_{k_m}+2 t_n)$ is a maximum, and factor out $\exp{[\beta_1 (t_{k_m}+2 t_n)]}$ from both numerator and denominator.