I've come to realize that 50% of numerical analysis mistakes that I make are due to overusing the approximation 1/(1-x) ~ 1-x

It always works for concentration inequalities, but burns you in fixed pt (its _okay-ish_ in float)

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But also, the idea is the following: if |x| < 1, 1/(1-x) = 1 + x + x^2 + …. So you can approximate 1/(1-x) by 1+x if x is small … but this is a numerically fraught approximation depending on your application (great for neural nets, horrible for eigenvalue estimation)

I don't understand this but seems legit so recasting in case there are other mathcasters

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You’re only missing a 1/sqrt(2*pi*n) though

Me personally, I use the approximation log(n!) = n log n - n. Somehow I always get things wrong with this…