I would like to approximate the following sum when $n \rightarrow \infty$ and $n \gg k$,

$$\sum_{x = k}^n \sum_{y > x}^n \frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose m}}.$$

The catch is that there is no closed form for partial sum of binomial coefficients. I wonder if there is a good way to approximate $$\frac{\sum_{m = 0}^{k - 1} {y - 2 \choose m}}{\sum_{m = 0}^k {y - 1 \choose m}}$$ so that the approximation converges to a value given that $n \rightarrow \infty$ and $n \gg k$. Any help is greatly appreciated.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.