Skip to main content

Questions tagged [riemann-sum]

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

Filter by
Sorted by
Tagged with
0 votes
1 answer
34 views

Showing integrability of f+g and additivity of the Darboux integral

I am currently working on the following question from Measure, Integration & Real Analysis by Sheldon Axler: Suppose $f,g:[a,b]\to\mathbb{R}$ are Riemann (Darboux) integrable on $[a,b]$. Prove ...
Alice's user avatar
  • 508
0 votes
0 answers
20 views

Use a Riemann sum to approximate the integral $\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot x}-1)dx$ in 1d

Consider the function $f:[-1,1]\setminus\{0\}\to \mathbb{R}$ given by $f(x)=\frac{1}{|x|^{2.5}}.$ For dimension $d=1,$ Consider the integral below: $$\int_{-1}^1\frac{1}{|x|^{2.5}}(e^{-i\pi\omega\cdot ...
Chang's user avatar
  • 329
4 votes
2 answers
259 views

Could we approximate $\int_0^1\frac{1}{x^4}dx$ using a Riemann sum?

We know that in one dimension, the integral $\int_{0}^{1}\left(1/x^{4}\right){\rm d}x$ is not finite. But could we approximate this integral using a Riemann Sum ?. ...
Chang's user avatar
  • 329
0 votes
0 answers
50 views

If every sequence of Riemann sums of a function converges, is the function integrable?

Let $f:[a,b]\to\mathbb{R}$ be a function, and $P_n$ the equidistance partition of $[a,b]$ into $n$ subintervals of an equal length. Let $P_n^\ast$ be the set of sample points from each subinterval of $...
ashpool's user avatar
  • 7,006
-1 votes
0 answers
37 views

What other ways can we derive a Riemann sum by thought experiment?

I was thinking about the traditional thought experiment that is used to derive the Riemann sums. I realized that Riemann sums can be derived with two other thought experiments (see 2 and 3). Are there ...
Jon's user avatar
  • 1
1 vote
1 answer
56 views

Riemann Sum vs Quadrature

Is it accurate to say that: Riemann sum leads to the following approximation methods: Left sum. Right sum. Midpoint sum. As Riemann sum uses the function value $f(x_i^*) \;$ where $\quad x_{i-1} \...
Ahmed's user avatar
  • 81
0 votes
1 answer
52 views

Working on details on the Secretary Problem [closed]

I've been trying to follow this proof of the optimal way to solve the secretary problem (ref. https://en.wikipedia.org/wiki/Secretary_problem). Everything is clear to me except where they are ...
Alex's user avatar
  • 142
0 votes
2 answers
77 views

Calculate this limit using integral

Giving $a$, $b$ are two positive numbers and $\alpha$ is a positive integer; calculate the limit: $$ \lim_{n\to\infty}\left[\dfrac{1}{n^{2\alpha+1}}\sum_{k=1}^{n}\left(k^2+ak+b\right)^{\alpha}\right] $...
Lê Trung Kiên's user avatar
0 votes
0 answers
17 views

Function-vector dualism of inner product

I have a question related to the dualism between the inner product of infinite-dimensional vectors and the integral over the product of their corresponding function representations, which is given by $...
Richard Schömig's user avatar
3 votes
1 answer
54 views

Conjecture: The $n$th left Riemann sum for $\int_0^1 (x-x^2)^k dx$ is $B(k+1,k+1) + \Theta(n^{-2 \lceil (k+1)/2\rceil})$

While playing around on Wolfram Alpha with integrals of the form $\int_0^1 (x-x^2)^k dx$, where it happens that Wolfram Alpha displays formulas for the $n$th left Riemann sum (using equally spaced ...
Daniel Schepler's user avatar
1 vote
2 answers
53 views

How to calculate the limit of this riemann sum using integrals?

$$ \lim_{n\to \infty} \frac{3}{n} \sum_{k=1}^{n} \sqrt{\frac{n}{n+3(k-1)}} $$ Assuming this can be written as a Riemann Sum, how can I bring it to an integral? I'm trying to make it reach the form $\...
Manar's user avatar
  • 375
3 votes
3 answers
73 views

Doubt regarding limits on riemann sums

Find $$\lim_{n\to\infty}\frac{1^p+3^p+\ldots+(2n+1)^p}{n^{p+1}}$$ I found a solution here which goes like this: By Riemann sums, for any $p>-1$: $$ \frac{1}{n}\sum_{k=0}^{n}\left(\frac{2k+1}{n}\...
math_learner's user avatar
1 vote
1 answer
80 views

A question about sum of sequence with fifth powers

$\sum_{r=1}^{p}(4p+3r)^5$ I'm looking for the coefficient of the highest degree term in the formula obtained when this sum is written in terms of $p$. Is there a practical way to do this? And also ...
Briston's user avatar
  • 192
0 votes
0 answers
36 views

Solving Sequence Using Riemann Sum

I am currently in my second calculus course and my professor asked me to evaluate the limit of a sequence. $$ b_k = \frac{1}{9k+1} + \frac{1}{9k+2} + \cdots + \frac{1}{20k} $$ We did a similar problem ...
Muhammad Ali Ullah's user avatar
0 votes
1 answer
79 views

Is this a Riemann sum?

I have come a cross with a sum that looks like this: $$\sum_{x\in{\Lambda_N}}\epsilon^2 k(\epsilon x)e^{-i\pi\omega \cdot \epsilon^2 x}\quad \quad\quad\quad(*)$$ Here $x$ takes values in the discrete ...
Chang's user avatar
  • 329
1 vote
0 answers
46 views

Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
SiegAndy's user avatar
0 votes
0 answers
25 views

Oscillation of an Integrable Function on a Subinterval is smaller than epsilon. Proof check.

Given an integrable function $f : [a,b] \rightarrow \mathbb{R}$, and a subinterval $[c,d] \subseteq [a,b]$ with $c < d$, we aim to prove that for every $\epsilon > 0$, there exists an interval $[...
mpavlov23's user avatar
  • 113
0 votes
0 answers
42 views

Is this Dirichlet type function Riemann Integrable?

\begin{cases} \cos\left(\frac{\pi}{x}\right) &,\quad \text{if } x \text{ is rational} \\ 0 &,\quad \text{if } x \text{ is irrational} \end{cases} Over the interval $[0,1]$ My approach Solve ...
Theorist's user avatar
0 votes
0 answers
38 views

A inequality of Darboux integral

As we know,if $f(x)$ are Riemann integrable,we have \begin{gather} \left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x. \end{gather} So,for Darboux integral,such as upper integral,...
Daeree's user avatar
  • 25
1 vote
1 answer
48 views

Is there a nice closed form of the following function: $f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+1}{n-2i}$

I am tring to find the closed from of the following function: $$f(k) = \lim_{n \rightarrow \infty}\prod^{kn}_{i = 0}\frac{n-2i+3}{n-2i+2},$$ where $k \in [0,\frac{1}{2}]$ If the numerator is $n-2i+4$ ...
0099ax43's user avatar
2 votes
0 answers
40 views

Evaluate $\lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{k}\tan(\frac{k\pi}{4n + 4})$

My approach: $\lim_{n\rightarrow\infty} \sum_{k=1}^n \frac{1}{k}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\rightarrow\infty} \sum_{k=1}^{n+1} \frac{1}{k}\frac{n+1}{n+1}\tan(\frac{k\pi}{4n + 4}) = \lim_{n\...
Moxy's user avatar
  • 329
0 votes
1 answer
66 views

How to prove every shell is non-overlapping in volume of revolution by "shells"? Does the Riemann sum imply the volume is over-counted?

Why is the shell method not $$\lim_{n \rightarrow \infty} \sum_{k=1}^n 2\pi\left(\frac{(b-a)}{n}\right)\cdot f\left((k-1)\cdot\frac{(b-a)}{n}\right)\cdot \frac{1}{n} + \pi f\left((k-1)\cdot\frac{(b-a)}...
user avatar
3 votes
1 answer
133 views

Evaluate $\int_0^3 x{\sqrt {3-x}}\;dx $ using Riemann sums

So, I was asked to do this integral using the limit method (or the Riemann Sum) $$\int_0^3 x{\sqrt {3-x}}\;dx $$ And, I do it like this: $$\int_0^3 x{\sqrt {3-x}}\;dx $$ Firstly, I determine the $\...
aki's user avatar
  • 61
0 votes
0 answers
96 views

Is $\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$ a valid definite integral riemann sum? What is it called if so?

I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha): $$\lim\limits_{n \...
riemannsumalt's user avatar
2 votes
0 answers
45 views

The behavior of $\lim_{n\to \infty}\sum_{k=1}^n \frac{1}{n+k+k^\alpha}, \alpha\in \mathbb R$.

As title, I am interested in the value of $$\lim_{n\to \infty}\sum_{k=1}^n\frac{1}{n+k+k^\alpha},$$ where $\alpha\in \mathbb R$. Here is my attempt: For $\alpha\in (0,1)$, note that $$\sum_{k=1}^n\...
SuperSupao's user avatar
1 vote
1 answer
50 views

Limit of $∞.0$ form of an integral and Riemann sum

I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$ where k is chosen such that the order ...
Cognoscenti's user avatar
0 votes
1 answer
47 views

Use Riemann sums to prove $\int_{1}^{b} \frac{1}{\sqrt{x}}dx = 2(\sqrt{b}-1)$ using equal subintervals

This post refers to Question 2 of the review problems at the end of Chapter 6 of George Simmon's Calculus: Following the general form $$\int_{a}^{b} f(x)dx = \lim \limits_{max \Delta x_k\to0} \sum_{k=...
RobinSparrow's user avatar
  • 2,042
5 votes
1 answer
184 views

Let $f:[0,1]\to \Bbb R$ such that $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$

A function $f$ is defined on $[0,1]$ by $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$ The solution given is as follows: $f$ is ...
Thomas Finley's user avatar
9 votes
1 answer
702 views

Calculating pretty difficult limit that invloves Riemann sums

Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit $$\lim_{n \to \infty} n\left(n\Big(\ln(1+\sqrt{2})-S_n\Big)-\frac{1}{2\sqrt{2}\,(1+\sqrt{2})}\right).$$ My intuition ...
Shthephathord23's user avatar
6 votes
2 answers
502 views

How to perform this sum

I encountered this sum $$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}} \frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$ ...
user824530's user avatar
3 votes
2 answers
107 views

Convergence of a sum as limit tends to infinity that seems to be harmonic series

I have come across a mathematical problem that is to evaluate the expression: $$ lim_{n\rightarrow\infty} \left\{\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+...+\frac{1}{\...
M.Riyan's user avatar
  • 124
0 votes
0 answers
122 views

Why is the following claim true in Exercise 7.4, Apostol's Mathematical Analysis?

On this page, a proof of the equivalence of two definitions of Riemann integrals is given by the user Pedro using Apostol's Hint for Exercise 7.4, Mathematical Analysis. However, I still find this ...
Gravitational Singularity's user avatar
0 votes
0 answers
42 views

Upper and Lower Integral [duplicate]

Given $F(x): [0,1] \rightarrow \mathbb{R}$. $F(x) = 0, x \in \mathbb{Q}$ and $F(x) \ge \frac{1}{2}, x \not \in \mathbb{Q}$. Proof/Disproof $F(x)$ Riemann-Integrable! My Attempt: Consider that, $$M_i = ...
Niccolo's user avatar
  • 694
1 vote
1 answer
72 views

How to prove if $f$ is Darboux integrable then for all $\epsilon > 0$ then $U(f, P_{\epsilon}) - L(f, P_{\epsilon}) < {\epsilon}$ ??

Background: I am studying Real Analysis (never studied it before) from the book 'Real Analysis' by Jay Cummings. I am at chapter 8 (Integration) when I encounter theorem 8.14 which comes almost right ...
Viraj Agarwal's user avatar
4 votes
1 answer
95 views

Compute Riemann Sum

I was not formally taught how to evaluate Riemann sums using the summation rule, and so I am going off of solutions to other problems to apply to my own problem. However, I am stuck. Any help would ...
Artemis F's user avatar
  • 173
0 votes
0 answers
140 views

Understanding the Definition of Indefinite Integral Using Riemann Sums

The definite integral of a function $f$ from $x=a$ to $x=b$ and $\Delta x = (b-a)/n$ is defined by the limit of a Riemann Sum: $$ \int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(a+i\cdot\...
Marco Moldenhauer's user avatar
1 vote
0 answers
55 views

Doubt about Rudin exercise 6.3 a

Define three functions $B_1, B_2, B_3$ as follows: $B_j(x) = 0$ if $x < 0$, $B_j(x) = 1$ if $x > O$ for $j = 1, 2, 3$; and $B_1(0) = 0, B_2(0) =1, B_3(0) = \frac{1}{2}$. Let $f$ be a bounded ...
pie's user avatar
  • 6,456
1 vote
1 answer
126 views

prove that $\int_{a}^{c} f(x) \,dx = \int_{a}^{b} f(x) \,dx + \int_{b}^{c} f(x) \,dx $

I've been working on a proof related to the additivity of Riemann integrals and would greatly appreciate insights and feedback for clarity and correctness of the proof. Because i've never seen a text, ...
rllynotgoodwithmath's user avatar
-1 votes
2 answers
85 views

Find a definite integral which represent $ \lim_{n\to\infty}{\frac{1}{n}\sum_{k=1}^{n}{\sqrt\frac{k}{n+k}}} $

Find a definite integral which represent $$\lim\limits_{n\to\infty}{\frac{1}{n}\sum\limits_{k=1}^{n}{\sqrt\frac{k}{n+k}}}.$$ I don't know how can I approach the question. Is the answer $$\int_{0}^{1}{\...
Overnight FYT's user avatar
0 votes
1 answer
50 views

Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$

Let $f, g$ be Riemann-integrable functions at $[0,1]$. Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$. Prove that $\...
X4J's user avatar
  • 1,052
0 votes
0 answers
33 views

Combination of function values on subinterval is bounded by $\inf$/$\sup$ of function.

Let $f:[a,b] \to \mathbb{R}$ be a real valued function. And $\mathcal{P} = (x_0, \ldots, x_n)$ be a partition of $[a,b]$. Why is $$\inf_{x\in[x_{k-1},x_{k}]}f(x) \leq f(x_{k-1})+\dfrac{f(x_k) - f(x_{k-...
spectre42's user avatar
  • 181
1 vote
1 answer
60 views

Where does the differential come from when passing from a discrete sum to a continuous case, for instance for the case of the inner product?

I'm trying to develop some intuition behind the inner product. I understand for two n-dimensional vectors we can compute the dot product as: $\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^{n} a_i b_i$ So ...
Thomas's user avatar
  • 11
0 votes
0 answers
95 views

Definite integral of ln(x) from 1 to a as the limit of a Riemann sum

For $a > 1$ determine the definite integral $$ \int_1^a ln(x) $$ as the limit of a Riemann sum. Hint: Use the partitioning $P_N = (x_0, x_1, …, x_N)$ with: $$ 1 = x_0 < x_1 < … < x_N = a \...
sagan's user avatar
  • 1
0 votes
0 answers
28 views

A question on exercise 10.6(b) in Munkres - Analysis on Manifolds

Let $$I=I_1\times\cdots\times I_n=[a_1,b_1]\times\cdots\times[a_n,b_n]$$ be a $n-$rectangle. Let $f:\ I\longrightarrow\mathbb R$ be bounded and $|f|\leq M$ on $I$. Let $P=P_1\times\cdots\times P_n$ be ...
PermQi's user avatar
  • 579
0 votes
1 answer
65 views

A question that was related to the evaluation about the errors between Riemann sum and double integrals

I have a question about the error evaluation between the double integral and its' riemann sums. It seems that this formula is apperantly not zero: $$\displaystyle\lim_{n\to\infty}n[\int_0^1 dx\int_0^1 ...
mumujun's user avatar
4 votes
1 answer
88 views

Evaluating $\lim_{n\to\infty}\sum_{k=0}^{n/2-1} \frac{k}{(n-k)^2} $

I am trying produce a closed form for $\lim_{n\to\infty}S_n$ where: $$ S_n = \sum_{k=0}^{n/2-1} \frac{k}{(n-k)^2} $$ For example, if $n = 10$: $$S_n = \sum_{k=0}^{4} \frac{k}{(10-k)^2} = \frac{0}{10^...
Michael's user avatar
  • 159
0 votes
0 answers
47 views

Help for Telescopic Riemann sum

Consider the Riemann sum $$\sum_{k=1}^n 2x^∗_k ∆x_k$$ of the integral of f(x) = 2x in an interval [a, b]. (a) Show that if $$x^∗_k$$ is the midpoint of the k−th subinterval, then the Riemann sum is ...
Gabrielle Santos's user avatar
2 votes
1 answer
65 views

Telescopic Riemann sum

Consider the Riemann sum $$\sum_{k=1}^n 2x^∗_k ∆x_k$$ of the integral of f(x) = 2x in an interval [a, b]. (a) Show that if $$x^∗_k$$ is the midpoint of the k−th subinterval, then the Riemann sum is ...
Gabrielle Santos's user avatar
3 votes
1 answer
127 views

Calculate $\int_0^1 x^2~dx$ only by using upper-/lower Darboux sums

Let be $f:[0,1]\to\mathbb{R}$ with $f(x)=x^2$. We know that $f$ is Riemann-integrable. Calculate $\int\limits_0^1 x^2~dx$ only by using upper-/lower Darboux sums. Let be $P$ a partition of $[0,1]$. ...
Philipp's user avatar
  • 4,564
0 votes
1 answer
85 views

Infimum of the Upper Sums [closed]

Let $P$ be the partition of the interval $[a,b]$. If $P$ is divided into two partitions $P_1$ and $P_2$ such that $P_1$ covers the interval $[a,c]$ and $P_2$ covers the interval $[c,b]$, then $U(P, f, ...
user avatar

1
2 3 4 5
30