These two approximations are given in my book. $$ \sum_{k=0}^m\left(\frac{k}{m}\right)^n\approx\int_0^m\left(\frac{x}{m}\right)^ndx\tag{1}$$
$$\sum_{i=m+1}^n\frac{1}{i-1}\approx\int_m^n\frac{1}{x}dx\tag{2}$$
I would like some explanation of why they are true and what does the "approximately" mean. Is it convergence in the limit or is it just close.
HINT
Write the RHS integral as a Riemann sum with the constant rectangle width. Pretty sure you intended $\int_0^m (x/m)^n dx$