I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which gives the upper bound $\sum_{k=0}^{r}\binom{n}{k} \leq \binom{n}{r}\frac{n-r+1}{n-2*r+1}$.
But for getting more precise approximation I was asked to use formula for fixed constant $t \in (0, 1)$ and $t^{-r}(1+t)^n$, where $t=\frac{n}{n-r}$.
I've transformed that statement $$t^{-r}(1+t)^n = t^{-r}\sum_{k=0}^n\binom{n}{k}t^k=t^{-r}\left(\sum_{k=0}^r\binom{n}{k}t^k + \sum_{k=r+1}^n\binom{n}{k}t^k\right)=\\=\left(\sum_{k=0}^r\binom{n}{k}t^{k-r} + \sum_{k=r+1}^n\binom{n}{k}t^{k-r}\right)$$ Because we need to approximate only result for $r$ I decided to through away the right sum $\sum_{k=r+1}^n\binom{n}{k}t^{k-r}$.
Now I have only $$\sum_{k=0}^r\binom{n}{k}t^{k-r}$$ which I should approximate by using the idea that $0 < t < 1$, but I have no idea how to do that.
Does anyone have any idea?