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Questions tagged [riemann-sum]

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

21
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2answers
531 views

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[...
17
votes
1answer
1k views

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}...
17
votes
0answers
352 views

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 $...
14
votes
1answer
10k views

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) ...
14
votes
2answers
850 views

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 $[a,b]$. $$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(...
13
votes
1answer
6k views

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?
12
votes
3answers
227 views

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: $$I=\lim_{n\to\infty}\dfrac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}...
12
votes
1answer
327 views

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 ...
11
votes
3answers
477 views

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 ...
11
votes
1answer
187 views

A way to solve the gap between prime numbers

Some time ago I found this sum of prime numbers converge(*) $$\sum_{k=1}^\infty \frac{p(k+1)-2p(k+2)+p(k+3)}{p(k)-p(k+1)+p(k+2)}\ \approx \frac{5}{7}\zeta(3)^{-2} $$ where: $\zeta(s)$ is the ...
10
votes
3answers
928 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
10
votes
3answers
4k views

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 ...
10
votes
1answer
1k views

Speed of convergence of a Riemann sum

Let $f(x)=x^d$ $(d\in(-1,0))$. We know that $$\sum_{i=1}^n \frac{1}{n}\left(\frac{i}{n}\right)^d\xrightarrow{n\rightarrow\infty}\int_0^1x^d dx=\frac{1}{d+1}.$$ My question is the following: Can we say ...
10
votes
2answers
631 views

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^...
9
votes
4answers
5k views

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 ...
8
votes
2answers
371 views

Perfect understanding of Riemann Sums

I am not sure I have completely and properly understood Riemann sums. Given a sum like: $$S_n = \displaystyle\sum_{r=1}^n \dfrac{r^4+ r^3n +r^2n^2 +2n^4}{n^5}$$ After dividing by $n^4$ we will ...
8
votes
2answers
221 views

Find $\lim\limits_{n\to\infty}{\left(1+\frac{1}{n^k}\right)\left(1+\frac{2}{n^k}\right)\cdots\left(1+\frac{n}{n^k}\right) }$ for $k=1, 2, 3, \cdots$

First of all, I already searched Google, math.stackexchange.com... I know $$ \lim_{n\rightarrow\infty} \left( 1+ \frac{1}{n} \right) ^n=e$$ That is $$ \lim_{n\rightarrow\infty} \...
8
votes
2answers
322 views

Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$

I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, ...
8
votes
1answer
263 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of $f$ ...
8
votes
0answers
241 views

Would an integral defined using partitions of an interval into infinitely many intervals make sense?

In the definition of Riemann integral or Darboux-integral we first study partitions (or tagged partition) of the given interval determined by finitely many points. To each partition and a function $f$ ...
7
votes
7answers
2k views

Finding limits using definite integrals $\lim_{n\to\infty}\sum^n_{k=1}\frac{k^{4}}{n^{5}}$

Find the limit of $\displaystyle\lim_{n\to\infty}\frac 1 {n^5}(1^4+2^4...+n^4)$ using definite integrals. It's equal to: $\displaystyle\lim_{n\to\infty} \sum^n_{i=1}\frac 1 i$ but now I'm not sure ...
7
votes
2answers
750 views

Infinite Series with Pi

I have this homework problem, that I'm stuck on. We know that:$$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ I have to find the sum of: $$\frac{1}{1^2}+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}...
7
votes
4answers
4k views

Prove that $f(x)=e^x$ is Riemann integrable using Riemann sums

Does anyone know how to prove that $f(x)=e^x$ is Riemann integrable using right or left hand Riemann sums?
7
votes
1answer
211 views

How to calculate the limit $\lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin(\frac{2k\pi }{n})$? Is it a Riemann sum?

I just came across this limit and I suppose it can be computed using a Riemann sum but I can't get it right. $$\lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin\left(\frac{2k\pi }{n}\right)$$ Any ...
7
votes
4answers
481 views

How to integrate $xe^x$ without using antiderivatives or integration by parts.

Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts. I tried it by ...
7
votes
1answer
486 views

Find $\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$ using Riemann sums

Find $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}}$$ using Riemann sums. I got $$\lim_{n \to \infty} n^2 \sum^n_{k=1} {1 \over {(n^2+k^2)^2}} = \lim_{n \to \infty} \sum^n_{k=1} ...
7
votes
4answers
471 views

Convergence of a sequence (possibly Riemann sum)

Let $a_1, a_2, a_3, . . . , a_n$ be the sequence defined by $$ a_n = 2\sqrt{n}-\sum_{k=1}^{n}\frac{1}{\sqrt{k}} = 2\sqrt{n} - \frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-...-\frac{1}{\sqrt{n}} $$ show ...
7
votes
2answers
392 views

Showing the Riemann integral is a linear operator over step functions

Given an interval $[a,b]$, let $S[a,b]$ be the set of all step functions for all possible finite partitions of $[a,b]$. Given a partition $x_0,\dots, x_n$ of $[a,b]$, define $\chi_i$ as the ...
7
votes
1answer
166 views

Showing limit of improper integral using Riemann sum

I can show that if $f$ is continuous for $x\ge0$ with $\lim\limits_{x \to \infty} f(x) = A$, then \begin{equation} \lim_{x \to \infty} \frac{1}{x} \int_0^x \! f(t) \, \mathrm{d}t = A \end{...
6
votes
3answers
802 views

Prove or refute: If $f$ is Riemann integrable on $[0,1]$, then so is $\sin(f)$

I'd love your help with the following question: I need to prove or refute the claim that for a Riemann integrable function $f$ in $[0,1]$ also $\sin(f)$ is integrable on $[0,1]$. My translation for ...
6
votes
5answers
360 views

Compute the following limit, possibly using a Riemann Sum

$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the ...
6
votes
2answers
367 views

A rigorous proof for a Riemann-like sum

I am asked to find the limit of the following sum: $$ \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\sin{(\frac{k}{n})}\sin{(\frac{k}{n^2})} $$ My attempt: For $n\rightarrow\infty$ and for $1\leq k\leq n$:...
6
votes
2answers
820 views

If $f$ is continuous on $[a,b)$ and $[b,c]$, then $f$ is Riemann integrable on $[a,c]$.

True or False: If $f$ is continuous on $[a, b)$ and on $[b, c]$, then $f$ is Riemann integrable on $[a, c]$. I was unsure if the $)$ in $[a,b)$ completely changed the problem and made it false and I ...
6
votes
2answers
1k views

Transforming a Riemann Sum to an integral.

Question. Consider the limit $$\begin{align} L&=\lim\limits_{n\to\infty}\sum_{k=1}^n\dfrac{k\sqrt{n+k}}{n^{5/2}}\\&=\lim\limits_{n\to\infty}\left(\dfrac{\sqrt{n+1}}{n^{5/2}}+\dfrac{2\sqrt{n+1}}...
6
votes
2answers
718 views

On the horizontal integration of the Lebesgue integral

I'm studying Lebesgue integral and its difference with respect to the Riemann one. I'm reading that the key difference (at least graphically speaking) is that the first slices the function ...
6
votes
3answers
663 views

Calculate an integral with Riemann sum

We know that Riemann sum gives us the following formula for a function $f\in C^1$: $$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f.$$ I am looking for an example where ...
6
votes
1answer
3k views

integration of 1/x as a riemann sum

To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: $$a, aq, aq^2, \ldots, aq^{n-1}, aq^n=b$$ where $q=\sqrt[n]{b/a}$. We then only need ...
6
votes
3answers
2k views

Relation between Right Riemann sum and definite integral

Let a partition $\{t_0,\ldots,t_n\}$ of the interval $[a,b]$ and let $f$ an integrable function. (we may also assume that $f$ is differentiable on $[a,b]$) We know that the Right Riemann sum is $$R(...
6
votes
1answer
1k views

A Riemann integrable function must have infinitely many points of continuity

I was wondering whether anyone would be so kind as to briefly check my proof? I am supposed to prove the statement without using any theorems which would render the proof trivial. If $\displaystyle\...
6
votes
1answer
661 views

Riemann Integrability in $\Bbb R^2$

Define the General Subdivision $S$ of a rectangle $R$ in $\Bbb R^2$ as a collection $E_1,...,E_k$ of Jordan regions such that none of them has interior points in common, and: $$R \subset \bigcup_{i=1}...
6
votes
2answers
123 views

$f(x)>0\implies\int_a^bf(x)\mathop{dx}>0$

Suppose $f$ is Riemann-integrable on $[a,b]$ such that $f(x)>0, \forall x \in [a,b].$ Prove: $$\int_a^bf(x)\mathop{dx}>0$$ Provided solution: Suppose by contradiction that $I=\int_a^bf(...
6
votes
0answers
263 views

Intuition of a tangent vector being a line segment in the context of integration

Following my previous question on the meaning of dx, and my read on A geometric approach of differential forms, I was trying to work out the intuition on paper. tldr of the post : $\frac{\partial f}{\...
6
votes
0answers
150 views

Understanding Stieltjes-Riemann

From my understanding from lectures, the Stieltjes-Riemann integral is a generalization of the Riemann integral. When using the identity function as integrator, the Riemann sum and Stieltjes-Riemann ...
6
votes
0answers
116 views

Infinite Sums which turn out to be Riemann Integrals

I'm looking for examples of infinite series which look hard to evaluate at first, but become very simple when viewed as a Riemann integral. An example would be $$\frac{1}{n+1}+\frac{1}{n+2}+ \ldots +\...
6
votes
0answers
102 views

Calculus II: Comparison Test

I have this math problem where I have to show that a sum converges. Is this correct? Thanks $$\sum_{n=1}^{\infty}\frac{2n-1}{ne^n}$$ I chose $\sum_{n=1}^{\infty}\frac{2n}{ne^n}$ to compare it to. ...
5
votes
3answers
172 views

evaluating $\lim_{n \to \infty }(\frac{n!}{n^n})^\frac{2n^4 + 1}{5n^5 + 1}$

I was trying to work out the following limit $$\lim_{n \to \infty }\Big(\frac{n!}{n^n}\Big)^{\Large\frac{2n^4+1}{5n^5+1}}$$ I have some idea about how to do it. Basically I expanded the $n!$ and ...
5
votes
4answers
479 views

Solve $\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$ without using Riemann sums.

The natural way (and I think, the easier) to solve $$\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$$ involves Riemann sums. Multiplying and dividing by $n$, we get $...
5
votes
3answers
179 views

limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$

How do I evaluate this? $$\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$$ I got concerned for that, I've tried make it integral for Riemann but it still undone.
5
votes
4answers
196 views

Evaluation of $\lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can write ...
5
votes
3answers
92 views

Need clarification in Riemann sum

I am a self-learner and the book that I am engaged in gave me some basics of Riemann sums but I see that they are not enough to solve most of the problems related to Riemann sums. Can someone give me ...