# Questions tagged [riemann-sum]

This tag is for questions about Riemann sums and Darboux sums.

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### Another series involving $\log (3)$

I will show that $$\sum_{n = 0}^\infty \left (\frac{1}{6n + 1} + \frac{1}{6n + 3} + \frac{1}{6n + 5} - \frac{1}{2n + 1} \right ) = \frac{1}{2} \log (3).$$ My question is can this result be shown ...
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### Is the definition of the Riemann sum from Thomas' Calculus correct?

I'm having trouble with theoretical understanding of the Riemann sum with this explanation/definition from Thomas' Calculus. I checked Wikipedia and it seems to state virtually the same.: On each ...
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### A filter in $\mathbb{R}$ by means of the Riemann sums

I trying to find a filter $\mathcal{F}$ in $\mathbb{R}$ defined by means of the Riemann sums. In the other words, let $f: [0,1]\subset \mathbb{R} \longrightarrow \mathbb{R}$ be a continuous function. ...
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### Is the sequence $x_n=\dfrac{1}{\sqrt{n}}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\ldots+\dfrac{1}{\sqrt{n}}\right)$ monotone?

Observe that $x_1=1$ and $x_2=\dfrac{1}{\sqrt{2}}\left(1+\dfrac{1}{\sqrt{2}}\right)>\dfrac{1}{\sqrt{2}}\left(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}\right)=1$. Thus, $x_2>x_1$. In general, we ...
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