Is there a closed form to $\sum\limits_{i = r}^{n}{ {{i}\choose{r}} x^i}$ ?
I'm familiar with the identity $\sum\limits_{ i = r}^{n} {{i}\choose{r}} = {{n+1}\choose{r+1}}$.
Context: I tried analyzing the expression $f(P) \equiv \lim\limits_{T\rightarrow \infty}\frac{1}{T}\sum\limits_{ i = 0}^{T}P^i$ for a stochastic matrix $P$ and also for the stochastic matrix $P' = (1-\epsilon)P+\epsilon I$. The claim is that $f(P)=f(P')$. While doodling a direct proof by using binomial coefficients I have encountered the expression in the title.