I recently came across this question:
Evaluate: $$\frac{\sum _{n=1}^m(n \times\log_2(n))}{m \times\log_2(m)}$$ However, I'm not certain where to begin, I considered finding the bounds to the equation by integration, but I don't know if there's any way which is more specific than that.
Thank you for your time!
As sum of logs: \begin{align} \frac{\sum _{n=1}^m(n \times\log_2(n))}{m \times\log_2(m)} &=\frac 1m\sum _{n=1}^m\frac{\log_2(n^n)}{\log_2(m)}\\ &=\frac 1m\sum _{n=1}^m\log_m(n^n)\\ &=\sum _{n=1}^m\log_m(n^{n/m}) \end{align} As product: \begin{align} \frac{\sum _{n=1}^m(n \times\log_2(n))}{m \times\log_2(m)} &=\frac 1m\sum _{n=1}^m\frac{\log_2(n^n)}{\log_2(m)}\\ &=\frac 1m\log_m\left(\prod_{n=1}^m n^n\right)\\ &=\log_m\sqrt[m]{\prod_{n=1}^m n^n} \end{align}
