UMVUE of $\frac{p}{1-p}$ when $X\sim bin(n,p)$ Uniform Minimum Variance Unbiased Estimate of $\frac{p}{1-p}$ when $X\sim bin(n,p)$
Note: $Bin(n,p)$ is an one parameter exponential family member with min complete sufficient statistic $X$. Then If can find $E[T(X)]=\frac{p}{1-p}$ Then by Scheffes  theorem then UMVUE, but will not achieve CRB because no linear function of X can produce an unbiased estimate of $\frac{p}{1-p}$. 
Then: 
$$E[T(X)]=\sum_{t=0}^{n}T(t){n\choose t}p^t(1-p)^{n-t}=\frac{p}{1-p}$$
$\Longrightarrow$
$$\sum_{t=0}^{n}T(t){n\choose t}p^{t-1}(1-p)^{n-(t-1)}=1$$
If we let $$T(t)=\frac{{n\choose t-1}}{{n\choose t}}$$ 
the desired equality follows then 
$$T(X)=\frac{{n\choose X-1}}{{n\choose X}}$$
is the UMVUE. Is my logic correct? Is there another way to find the UMVUE? 

Unfortunately your estimator isn’t an unbiased estimator for $\frac p{1-p}$. We have
$$
\sum_{t=1}^{n+1}\binom n{t-1}p^{t-1}(1-p)^{n-(t-1)}=(p+(1-p))^n=1\;,
$$
but your sum runs from $0$ to $n$. You could fix the problem at $0$ by choosing $T(0)=0$, but you can't make up for the missing term for $t=n+1$. This is the only term that doesn’t go to zero for $p\to1$, and the fact that it’s missing reflects the fact that you can’t mimick the $\frac1{1-p}$ divergence with a polynomial in $p$ on the left-hand side.
