A Riemann sum I

Evaluate the limit of the sequence

$\ell = \lim_{n\to\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n}\right)$

Solution

We simply note that

$\begin{align*} \frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n} &= \frac{1}{n} \sum_{k=1}^{2n} \frac{1}{1+\frac{k}{n}} \\ &\rightarrow \int_{0}^{2} \frac{\mathrm{d}x}{1+x} \\ &= \ln 3 \end{align*}$

and that’s all folks!

On the zero function

A Riemann sum I

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