Calculate the Riemann's sum

Question

Calculate the Riemann's sum of $\:\: \displaystyle \mathbf{f(x)=\frac{1}{x+2}}\:\:$ in $\:\:\mathbf{[1,2]} \:\:$ using $\mathbf{P \in \mathbf{[1,2]}} \:\:$ uniform partition. I need help to finished the sum, because $\left(with \:\: \mathbf{x_k=1 + hk}, \displaystyle \mathbf{h=\frac{1}{n}} \right)$ $$\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{1+hk+2}h=\lim_{n \rightarrow \infty} \sum_{k=1}^{n} \frac{1}{3n+k}$$ Exists other method with uniform P? I don't know how to resolve this sum

Answer 1

I'm not sure if this is what you are expected to do here but

$$ \sum_{k=1}^{n} \frac{1}{3n+k}=\sum_{j=3n+1}^{4n} \frac{1}{j}=H(4n)-H(3n)$$

where $H(n)$ is the Harmonic number. Then you can use the expansion $H(n)=\log n + \gamma + O(1/n)$.

(Though you normally work the other way round, you use the integral to prove the expansion).