Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–Laplace theorem with Stirling's formula. To make it concrete: Let $B_n$ be binomially distributed and let $N$ have the standardized normal distribution. I want to find an upper bound for $$\epsilon_n = \sup_{a<b} \left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$$

I want to compare this error with the best known error estimation of the Berry-Essee theorem for the binomial distribution.

My Question: Do you know any proof in a textbook / paper / article where the theorem by De Moivre and Laplace is proved with Stirling's formula and the total error is estimated?

The proof shall not only show the speed of convergence but also estimate the value of any occurring constant in the error estimation. Can you point me to this proof, please?

The results of my inquiry so far: I have only found a proof which shows the convergence speed $\epsilon_n \in O\left(\frac 1{\sqrt n}\right)$. See this proof by Márton Balázs and Bálint Tóth (which also just considered $\left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$ without the supremum). Other proofs do not investigate the error at all (see for example this proof on Wikipedia).

I also asked this question on MSE. Unfortunately I only got a reference to a proof, where the characteristic function is used (I look for a proof which uses Stirling's formula and not the characteristic function). That's why I reask this question here. I hope, that's okay... Thanks for your answers!

Context for my question: For one part of my thesis I try to find an upper bound for the error in the normal approximation of the binomial distribution following the standard proof of the De Moivre–Laplace theorem with Stirling's formula. To make it concrete: Let $B_n$ be binomially distributed and let $N$ have the standardized normal distribution. I want to find an upper bound for $$\epsilon_n = \sup_{a<b} \left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$$

I want to compare this error with the best known error estimation of the Berry-Essee theorem for the binomial distribution.

My Question: Do you know any proof in a textbook / paper / article where the theorem by De Moivre and Laplace is proved with Stirling's formula and the total error is estimated?

The proof shall not only show the speed of convergence but also estimate the value of any occurring constant in the error estimation. Can you point me to this proof, please?

The results of my inquiry so far: I have only found a proof which shows the convergence speed $\epsilon_n \in O\left(\frac 1{\sqrt n}\right)$. See this proof by Márton Balázs and Bálint Tóth (which also just considered $\left|\mathcal P\left(a\le \frac{B_n-np}{\sqrt{np(1-p)}} \le b\right)-\mathcal P(a \le N \le b)\right|$ without the supremum). Other proofs do not investigate the error at all (see for example this proof on Wikipedia).

I also asked this question on MSE. Unfortunately I only got a reference to a proof, where the characteristic function is used (I look for a proof which uses Stirling's formula and not the characteristic function). That's why I reask this question here. I hope, that's okay... Thanks for your answers!