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

  • $\begingroup$ I think you've done everything you can. There is no closed form for such a sum. At least none of the answerers here would know of one. Quite the opposite actually. Our ability to calculate this integral by other means is the tool for finding this limit. My guess is that whoever gave this exercise either wanted you to do just this or, had in mind a fixed value of $n$, and just wanted to make you form a Riemann sum, and calculate it to approximate the integral. $\endgroup$ – Jyrki Lahtonen Aug 22 '17 at 4:29

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).

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