# Questions tagged [riemann-sum]

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

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### Calculate limit of a sum as an integral [closed]

I need to calculate this limit as a definite integral but it doesn't look like Riemann sum at all: $$\lim_{x\to\infty} n^2 \sum_{i=1}^{n} \frac{1}{(n + i + 1)^3}$$ What would be an approach to this?...
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### Fake proof that all functions are integrable [duplicate]

Below I present a proof that I know is wrong. It "states" that all bounded functions are integrable. However, I am not sure why it is wrong, i.e., I am unsure where my logic fails. It goes ...
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### Proof Check: Equivalence of Riemann and Darboux Integrals

I'm practicing my $\delta$-$\epsilon$ proofs by verifying that the Riemann and Darboux integrals are equivalent for functions on a closed interval. Here are the definitions I'm working with (taken ...
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### Summation bound above integral

How is this true? $\frac{\sqrt{k}}{k^2}+\frac{\sqrt{k+1}}{(k+1)^2}+\cdots+\frac{\sqrt{n}}{n^2} \leqslant \int_{k-\frac{1}{2}}^{\infty} \frac{\sqrt{x}}{x^2} d x$ where $k,n$ are natural numbers. I can ...
Let $a$ and $b$ be real numbers, such that $a < b$. We say $\Delta$ is a division of the interval $\left[a, b\right]$, if $\Delta=(x_0, x_1, x_2, ..., x_n)$, for some non-zero natural number $n$, ...
I have the sum \begin{align*} \sum_{j=1}^T\frac{1}{h}g\left(\frac{\omega_j-\omega}{h}\right)f(\omega_j),\quad \omega_j = \frac{2\pi j}{T} \end{align*} where $h>0$ and $\omega \in[0,\pi]$. I'm asked ...