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Questions tagged [riemann-sum]

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

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Working on details on the Secretary Problem

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
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2 answers
74 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
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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
52 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
52 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
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How can I calculate the limit of this riemann sum using integrals? [duplicate]

I have been trying to force this limit for hours now $\lim_{n\to \infty} (\frac{n}{n^2 + 1} + \frac{n}{n^2+4}+...+ \frac{n}{n^2+n^2})$ Assuming it can be written as a Riemann Sum $\lim_{n\to \infty} \...
Manar's user avatar
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3 votes
3 answers
71 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
71 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
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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
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1 answer
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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
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1 answer
45 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
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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
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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
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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
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1 answer
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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
39 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\...
Marin's user avatar
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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
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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
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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
44 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
48 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
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1 answer
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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
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5 votes
1 answer
172 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
697 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
492 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
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0 answers
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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
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0 answers
40 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
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1 vote
1 answer
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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
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0 answers
116 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
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1 vote
1 answer
106 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
83 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
48 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
  • 1,060
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
89 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
26 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
60 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
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0 answers
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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
64 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
119 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
83 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 vote
0 answers
46 views

Theorem on Riemann integral of monotone functions

I want to prove the following: Theorem: Let $(f_n)$ be a sequence of monotone (integrable) functions $f_n : [0,\infty) \rightarrow \mathbb{R}$ (for $t_i \leq t_j$ we have $f_n(t_i) \leq f_n(t_j)$ such ...
user57's user avatar
  • 760
1 vote
0 answers
50 views

Area under a curve using the rectangle method

Take $$x_k$$ as the left endpoint of each subinterval to find the area under the curve y = f(x) above the specified interval. f(x) = 9 − x^2; [0, 3] What I've done so far is to consider $$\Delta x = \...
Vitoria Santos's user avatar
0 votes
1 answer
61 views

Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable

Suppose $f : [0,1] \to \mathbb R$ be such that $$\displaystyle\sum_{i=1}^n |f(t_i) - f(t_{i-1})|^2 < 100$$ for any $n \in \mathbb N$ and $0 \leq t_0 < \cdots < t_n \leq 1$. Prove that $f$ is ...
Squirrel-Power's user avatar
1 vote
2 answers
125 views

Improper integral convergence implies the existence of an infinite series which its partial sum converges

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a Riemann-integrable function at $[0, \beta]$ for each $\beta \in (0, \infty)$. Suppose that $\forall x \in [0, \infty): \space f(x) \geq 0$. If the ...
X4J's user avatar
  • 1,060
1 vote
1 answer
239 views

Upper and Lower Sums for Negative Functions

The textbook I'm using to study integral calculus usually assumes for it's proofs that the function takes on only positive values. The author says that if we divide the x-axis into intervals, and pick ...
Camelot823's user avatar
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