Suppose I have a probabilistic Turing machine for a language such that$$ x \in A \implies \Pr(M \text{ accepts } x) \ge \frac{1}{n}$$$$ x \notin A \implies \Pr(M \text{ accepts } x) = 0$$I want to show that the above language is in $\mathsf{RP}$. In particular, I need a probabilistic Turing machine $N$ such that $$ x \in A \implies \Pr(N \text{ accepts } x) \ge 1 - \frac{1}{2^n}$$ where $n = |x|$.I tried to construct $N$ such that it runs $M$ multiple times (say $t$ times) and accepts if at least one of the runs accepts. This preserves the one-sided error but I'm not sure if it runs in polynomial time.  I showed that $x \in A \implies \Pr(N \text{ accepts } x) \ge 1 - (1 - \frac{1}{n})^t$.  Now, to show that it is in $\mathsf{RP}$, $$1 - (1 - \frac{1}{n})^t \ge 1 - \frac{1}{2^n}$$$$ (1-\frac{1}{n})^t \le \frac{1}{2^n} $$I am basically stuck at this point and unable to show an upper bound for $t$. I know that it suffices to show that $t$ is polynomial in $n$ but I am not able to prove it. How do I show that $N$ is indeed a polynomial time machine? Or maybe this is not the right approach?
Improving success probability for one sided error algorithms
complexity theory;randomized algorithms