Skip to main content

Approximation

In Probability,

ln⁡n!=ln⁡∏k=1nk=∑k−1nln⁡k≈∫x=1nln⁡x dx\ln n! = \ln \prod\limits_{k=1}^n k = \sum\limits_{k-1}^n \ln k \approx \int_{x=1}^n \ln x ~dx

Using integration by parts

∫u dv=uv−∫v du\int u ~dv = uv - \int v~du
d(uv)=v du+u dvd(uv) = v~du + u~dv
u=ln⁡x, dv=1, du=1x, v=xu = \ln x,~ dv = 1,~ du = {1 \over x},~v=x
∫x=1nln⁡x dx=[xln⁡x−∫x1x dx]x=1n\int_{x=1}^n \ln x~dx = [{x \ln x - \int x {1 \over x} ~dx}]^n_{x=1}
=[xln⁡x−x]x=1n=nln⁡n−n+1= [x \ln x - x ]^n_{x=1} = n \ln n - n + 1