$\lim_{n \to \infty}(1/(n^3+1)+1/(n^3+2)+...+1/5n^3)=\ln5$

Question

How do I show that $$\lim_{n \to \infty}(1/(n^3+1)+1/(n^3+2)+...+1/5n^3)=\ln 5$$ I know this can be done using an integral but for this particular question I cannot simply find an equivalent Riemann sum.Any answers will be welcomed.

Answer 1

Use an increment of $\frac1{n^3}$ $$ \begin{align} \sum_{k=1}^{4n^3}\frac1{n^3+k} &=\sum_{k=1}^{4n^3}\frac{1/n^3}{1+k/n^3}\\ &\sim\int_0^4\frac{\mathrm{d}x}{1+x} \end{align} $$ where $x\sim k/n^3$ and $\mathrm{d}x\sim1/n^3$