The Problem
Here is a part of the constraint of a linear programming. $$ \begin{align} \displaystyle\sum_{n=0}^{\infty}\pi_n=1,\;\pi_n&> 0,\forall n\geq0.,\\ \displaystyle 0<a\leq \frac{\pi_{n+1}}{\pi_n}\leq b&<1,\forall n\geq0. \end{align} $$
As we can see, $\pi_n$ is a probability distribution. $\pi_n$ is strictly decreasing, and the decrease speed has an upper bound $b$ and a lower bound $a$.
What I want to do is construct a new feasible solution from an old one. The new solution is decreasing at the slowest speed, that is $b$, before a constant $N^*\in N$. Formally, given any feasible solution $\{\pi_n,n\geq 0\}$, the aim is to construct a new feasible solution like
$$\widetilde\pi_n=\left\{ \begin{align} &\square\pi_0b^n, &0\leq n\leq N^*,\\ &\square, &n> N^*. \end{align} \right.$$
My Idea
My idea is straight. Just use a normalization constant $Z$ as follows,
$$\widetilde\pi_n=\left\{ \begin{align} &\frac{1}{Z}\pi_0b^n, &0\leq n\leq N^*,\\ &\frac{1}{Z}\pi_n, &n> N^*. \end{align} \right.$$ where $Z=\sum_{n=0}^{N^*}\pi_0b^n+\sum_{n=N^*+1}^{\infty}\pi_n$. But $\widetilde\pi_n$ doesn't add up at $N^*$.
Can anyone give me a feasible solution or some hints? Thanks!