Questions tagged [riemann-sum]

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

The Riemann sum is an approximation that is calculated by dividing the region you are working in into shapes. These shapes form a smaller region (similar to the one you are measuring) and then calculating the area of these smaller shapes. Then you add all these small areas together to give the approximation.

It was considered the foundation of integration until the introduction of the much more rigorous Lebesgue integral in 1904.

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Speed of convergence of Riemann sums

This question is inspired by a previous question. It was shown that, for all function $f \in \mathcal{C} ([0, 1])$, $$ \lim_{n \to + \infty} \sum_{k=0}^{n} f \left( \frac{k}{n+1} \right) - \sum_{k=0}^{n-1} f \left( \frac{k}{n} \right) = \int_0^1 f…
D. Thomine
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Compute $\lim\limits_{n\to\infty}(x_{n+1}-x_n)$ if $x_n =\sum\limits_{k=1}^{n-1}f(\frac kn)$ and $f$ continuous (but not continuously differentiable)

The following question from Furdui's book (Exercise 1.32. page 6) is an "open problem" : Let $f: [0,1] \to \mathbb{R}$ be a continuous (and not a continuously differentiable) function and let $$x_n = f\left(\dfrac{1}{n}\right) +…
Emma
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Does this sum of prime numbers converge?

$\newcommand{\P}{\operatorname{P}}$I'm wondering if this sum of prime numbers converges and how can I estimate the value of convergence. $$\sum_{k=1}^\infty \frac{\P[k+1]-2\P[k+2]+\P[k+3]}{\P[k]-\P[k+1]+\P[k+2]}$$ $ \…
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If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
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General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two integrals are the same. However, I want to know as…
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approximation of integral of $|\cos x|^p$

Let $p\in [1,2)$. Let $$ \beta = \frac{1}{2\pi}\int_0^{2\pi} |\cos x|^p\, dx = \frac{\Gamma(\frac{p+1}{2})}{\sqrt{\pi}\Gamma(1+\frac{p}{2})}. $$ Consider the following approximation to the integral definition of $\beta$: $$ S_n =…
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The absolute value of a Riemann integrable function is Riemann integrable.

This is an exercise in Bartle & Sherbert's Introduction to Real Analysis second edition. They ask to show that if $I=[a,b]$ is a closed bounded interval and that $f:I\to\mathbb{R}$ is (Riemann) integrable on $I$, then $|f|$ is integrable on $I$. Of…
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Is the sum of a Darboux function and a continuous function Darboux?

A Darboux function is a function that has the intermediate value property. That is a function $f$ such that $$ \forall a,b \in \mathbb{R} : f[a,b] \supseteq [f(a),f(b)] \cup[f(b),f(a)] $$ We define the sum of two functions as such $$ (f+g)(x) =…
Sriotchilism O'Zaic
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Is the indicator function of the rationals Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no matter what partition I choose, there will always be rational…
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Help with proving a statement based on Riemann sums?

Suppose we have the original Riemann sum with no removed partitions, where $f(x)$ is continuous and reimmen integratable on the closed interval…
Arbuja
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Evaluating some Limits as Riemann sums.

I really have difficulties with Riemann Sums, especially the ones as below: $$\lim_{n\to\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{3n}\right)$$ When i try to write this as a sum, it becomes $$\frac { 1 }{ n } \sum _{ k=1 }^{ 2n }…
Hckr
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Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in person or within a text, the discussion sort of…
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How prove this limit $\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{i+j}{i^2+j^2}=\frac{\pi}{2}+\ln{2}$

show that: this limit $$I=\lim_{n\to\infty}\dfrac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}\dfrac{i+j}{i^2+j^2}=\dfrac{\pi}{2}+\ln{2}$$ My try:…
user94270
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Closed form of $\sum^{\infty}_{n=1} \dfrac{1}{n^a{(n+1)}^a}$ where $a$ is a positive integer

Recently, I bought a book about arithmetic. I saw a question is like that: Given that $\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\cdots=\dfrac{\pi^2}{6}$, find the value of $$\dfrac{1}{1^32^3}+\dfrac{1}{2^33^3}+\dfrac{1}{3^34^3}+\cdots$$ Then,…
MafPrivate
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Riemann sum on infinite interval

It is well known that in the case of a finite interval $[0,1]$ with a partition of equal size $1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)=\int_0^1 f(x)dx$$ I was wondering under which conditions…
user223935
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