Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 7 / a / Solution

The logarithm turns the product defining the factorial into $\ln (N!)={\sum }_{k=2}^N\ln k$. Since $\ln x$ is increasing, each term bounds integrals on its adjacent unit intervals:Summing from $k=2$ to $N$ proves the integral bounds for a logarithmic factorial:For $N=1$, both sides and the empty sum are zero. For $N\ge 2$, the rectangle diagram gives the same inequalities geometrically. These bounds establish the leading $N\ln N-N$ growth, while Stirling's formula gives finer corrections.