What are some general tips/tricks for, in practice, approximating discrete things with high number of terms with continuous ones?
To motivate this question, i was trying to arrive myself at an approximation of $P(X \in [a,b])$ in the binomial distribution, with high number of trials, which i could not find a simple formula for. But i have no idea what to do. I am aware that it can be approximated by the normal distribution, which i purposely did not look at since i want to understand how i could arrive at it myself. But i just stare at the sum $\sum_{k= \lceil a \rceil}^{\lfloor b\rfloor} \binom nk p^k(1-p)^{n-k}$ without any idea what to do...