\frac{n^p-\frac{n^{p+1}-(n-1)^{p+1}}{p+1}}{n^p-(n-1)^p}=\frac{1-\frac n{p+1}\left(1-\left(1-\frac 1n\right)^{p+1}\right)}{1-\left(1-\frac 1n\right)^p}=\frac{1-\frac n{p+1}\left(\frac{p+1}n-\frac{p(p+1)}{2n^2}+O(n^{-3})\right)}{\frac pn+O(n^{-2})}=\frac{\frac p2+O(n^{-1})}{p+O(n^{-1})}