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Questions tagged [riemann-sum]

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

17
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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 $...
8
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0answers
242 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 $f$ ...
6
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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
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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
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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
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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
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0answers
175 views

Approximation of a Riemann sum (not really) by a Laplacian integral

I have a sum of the form: $$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$ where $f(x)$ and $g(x)$ are smooth functions defined for $0\le x \le 1$. I am interested in the Asymptotic ...
5
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2k views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
4
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0answers
80 views

Arc length of a Polar curve as a Riemann sum

Suppose we have a curve in polar plane satisfying the equation $r=f(\theta)$ with $\theta\in[a,b].$ To find the area enclosed by this curve in this range of $\theta$ using Riemann integrals, we ...
4
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184 views

Textbook Definition of Riemann Integral: Supremum, Infimum, Lower/Upper Bounds

The following excerpt is from my Laplace transform textbook, where the author is trying to define the Riemann integral: Let $F(x)$ be a function which is defined and is bounded in the interval $a \...
3
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29 views

Need Help Solving Or Finding The Solution To The Following Darboux System Of Nonlinear Equations.

I am working on a personal math project of mine and in order for me to continue I need to know the solution to this following system of nonlinear equations I am attaching as a photo. This equation ...
3
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67 views

Newman's proof of the Asymptotic Formula for the Partition Function

I'm working on Donald J. Newman's proof that $p(n) \sim \frac{1}{4\sqrt{3}n}e^{\pi\sqrt{\frac{2n}{3}}}$, as found in Chapter II of his book Analytic Number Theory. Here's what we have so far: the ...
3
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0answers
163 views

Riemann Sum, Ergodic Theorem and Strong Law of Large Number

Suppose $\eta_t$ is a pathwise Riemann integrable stochastic process, with $t\in[0,1]$. Consider, for every partition $\{t_{j,n}=\frac{j}{n}|~j=0,...,n\}$ of $[0,1]$, the Riemann sum $$ S_n=\sum_{j=0}...
3
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0answers
135 views

Converting a series to a Riemann sum,

I am manipulating a series and have gotten this far: $$ \lim_{n\to\infty} \sum_{m=1}^n \frac {(\frac{m}{n})^{p-1}}{1+ (\frac{m}{n})^p} \frac{1}{n}$$ I want to now say that this is a Riemann sum, ...
3
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0answers
80 views

How to prove that : $\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
3
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0answers
661 views

How to show that a piecewise constant function is integrable, using the upper and lower sums?

Let $f(x) = \begin{cases} 1 &\mbox{if } 0\leq x<1 \\ 3 &\mbox{if } 1\leq x<2 \\ 2 &\mbox{if } 2\leq x\leq 3. \end{cases}$ Show that $f(x)$ is integrable by $(a)$ ...
3
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1k views

Show $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b]

Let $f$ be bounded on a nondegenerate interval $[a,b]$. Prove that $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b] such that P is a ...
3
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48 views

Check my proof: Prove that if $f$ is defined as having a positive disntinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable

Prove that if $f$ is defined as having a positive discontinuity at $c$ and $0$ otherwise on [a,b], it is Darboux integrable and its integral is 0. $\forall \epsilon>0,$ choose $\delta=\frac{\...
3
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0answers
2k views

Finding the upper and lower Riemann-sums of trigonometric functions

I am asked to find Riemann-sums for the function $f(x) = \cos x, x \in [0, 2\pi], n = 4 \rightarrow \Delta x = \frac{\pi}{2}$ I was able to get the correct answers by intuition, but I have some ...
2
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0answers
10 views

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions? I know that by the definition of an Archmedean sequence of ...
2
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0answers
48 views

Convergence of sum to integral

I would like to estimate the absolute value of the following difference $$ \Delta(L) = \sum_{\alpha=-L+1}^L \frac{1}{1+2 L} e^{i t \sec^2\left(\pi\frac{\alpha - 1/2}{2 L+1}\right)} - \int_{-\frac{1}{...
2
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0answers
30 views

Rate of convergence of a Riemann sums for functions with poles

Let $f$ be a function $C^\infty(\mathbb{R}^d)$ and bounded. Moreover, suppose that $f \asymp |x|^k, k \ge 1$ as $x \to 0$. What can I say about the speed of convergence of Riemann sums of $\frac{f(x)}{...
2
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0answers
125 views

Limits in definite integral and Riemann sum

The relation between Riemann sum and definite integral is as below. $$\lim_{n\to \infty} \sum_{k=1}^n f(c_k) \, \Delta x = \int_a^b f(x) \, dx$$ How to determine the interval $(a,b)$ if only the ...
2
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0answers
223 views

Every monotonic function on $[a,b]$ is integrable?

My book says that the statement in the title is true. What is so special about a function being monotonic? When I think of a non (Riemann) integrable function, I think of $f$, where $f$ is $1$ on ...
2
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0answers
72 views

Proving that the existence of given limit implies that bounded $f:[a,b]\to\mathbf R$ is Riemann integrable

I'm struggling a bit with this problem I've been given. It regards Riemann integrability for a function $f$. It reads like this: Decide whether the statement is true or false by giving a proof if ...
2
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0answers
37 views

Define $f(x)=0$ if $x\in \mathbb{Q}^c$; $f(x)=p-q$ if $x\in \mathbb{Q}$ where $x=\frac{p}{q}$ in lowest terms

Define $f(x)=0$ if $x\in \mathbb{Q}^c$; $f(x)=p-q$ if $x\in \mathbb{Q}$ where $x=\frac{p}{q}$ in lowest terms (conventional way to represent rationals). The question asks that whether f is Riemann ...
2
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0answers
737 views

Proof of Riemann integral as limit of Riemann integral sum

I want to Prove the following statement, I will be appreciate if some one help me to do that. Let $f:[a,b]\to R$ and $f$ is bounded, show that if $f \in R$ ( Riemann integrable) and $\int_a^b ...
2
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0answers
72 views

Computing an integral in $\mathbb{R}^2$ by definition

Given the function $f(x,y) = xy$, how do I calculate $$\int_{[0,1]\times [0,1]}f(x,y)$$ by the definition of the Riemann integral, without showing it is Riemann integrable, since this comes from ...
2
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0answers
158 views

Find the general formula for summation of square root of rational function

We were given a problem statement saying: Using discrete sum, find the area of the unbounded region limited by the curve $y^2=\frac{x(x-3)^2}{6-x} , x\ge3$ and its asymptote. I've made the following ...
2
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0answers
343 views

Uniform Probability and Riemann Sum

Quoted from the Wikipedia page about Natural Density: We see that this notion can be understood as a kind of probability of choosing a number, which obviously is the reason why Natural Densities are ...
2
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0answers
120 views

Why does the following Riemann integral exist, but the other doesn't?

By definition if the upper integral equals the lower integral, then $ f $ is Riemann integrable. An example of a Riemann integrable function is $ f(x)=0 $ if $ x\in(0,1] $ and $ f(x)=1 $ if $ x=0 $. ...
2
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0answers
158 views

Bochner integral vs regulated integral

I'm reading Serge Lang's Real And Functional Analysis and at some point he introduces the regulated integral in order to prove the Fundamental Theorem Of Calculus (in the context of Banach Spaces), or ...
2
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0answers
626 views

Difference between lower sums and lower integral

Why is it true that the lower sums of f with respect to some partition is less than the lower integral (which is the supremum of the lower sums) I think what I'm confused about is the difference ...
2
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0answers
60 views

A problem on Riemann integration

I am stuck in proving that $$\sup_P L(P,f+g) = \sup_P L(P,f) + \sup_P L(P,g)$$. where $f$ and $g$ are Riemann integrable. How to do ?
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64 views

Is this kind of limit defined?

Is there any limit like this $$\lim_{f(x)\to0} g(x)=0?$$ Defined as follows$$\forall \epsilon>0,\exists \delta>0 | |g(x)|<\epsilon ,\forall x |0<|f(x)|<\delta?$$ Where f ...
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0answers
39 views

Counter-Example for Darboux Sums: “Finer” Partition with Greater Difference.

Let $P = \{p_i\}_{i= 1, n}$, $P' = \{p'_j\}_{j = 1, m}$ be partitions of an interval with max$|p'_j| \le $min$|p_i|$, i.e. all the sub-intervals of $P'$ are at least as short as all the sub-intervals ...
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0answers
24 views

Darboux sums elementary question - am I correct

I'm very new to this material and I would like someone more experienced to give input if possible. $Q \subset \mathbb R^n$ is a box and $f: Q \to \mathbb R$ is a function. Let $\Xi_Q$ be the set of ...
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0answers
31 views

Solving the integral of a step function

I am dealing with a step function S(t). The true functional form is not given or unknown, but what is known is that S(t) takes a different value at each time point t like this below. ...
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0answers
32 views

converting a regular summation to a riemann summation

I'm trying to convert the regular summation $$\sum_{i=1}^{30}3i+1$$into a Reimann sum. (and eventually into an integral) The issue, however, is that I can't find a way to format the summation in a way ...
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0answers
56 views

Riemann Sum - correct formula for an integral?

Which one of these is correct, are both of these right? $$\Delta x_i = (x_i-x_{i-1}) $$ $$ \int_a^b f(x) \, dx = \lim_{\|x\| \rightarrow 0} \ \sum_{i=1}^n f(x_i)\Delta x_i $$ $$ \int_{x_0}^{x_n} f(...
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0answers
139 views

How to calculate Upper/Low sums of an arbitrary partition?

If I'm given a function $f$ that is continuous on $[a,b]$ and asked to find the Upper sum $U(f,P)$ and Lower sum $L(f,P)$ where $P$ is an arbitrary partition of of a given interval $I$; $P$ = $\{x_i\}...
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0answers
123 views

How does double Riemann sum actually work?

I'm in an advanced calculus class and studying double integral. My question is about how double Riemann sum actually work as algebraic steps? I mean I understand essentially it is summing up the ...
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0answers
64 views

$\Delta x$ in the limit-definition, or Riemann-Sum-definition, of an Integral.

Every time I evaluate an integral as a Riemann Sum and I see $\Delta x$ as $\left(1/n\right)$, I think of an upper limit that could be $a+1$ and a lower limit that could be $a$, where $a$ is any real ...
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45 views

Theorem about estimating the error in definite integral approximation

Let $f(x)$ be a differentiable function on [$a,b$] and $T$ partition of [$a,b$], $T=${$x_0,x_1,\ldots,x_n$}. For arbitrary points $u_i$ we want to calculate the difference: $$\left\lvert{\int_a^b{f(x)...
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0answers
58 views

Evaluating an integral using Riemann sums

I just started learning this topic and I'm quite confused about the methods that are being used to create the riemann sums. The method I tried using involves calculating the integral as the sum: $$\...
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0answers
67 views

Expectation of Riemann sum with integral inside

I am trying to solve the following thing. I have two random processes $(X_t)$ and $(Y_t)$, on the interval $[0,T]$. For what is worth, $(X_t)$ is non-negative. I want to show that $$ \lim_{\Delta_t\...
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0answers
90 views

Control speed of convergence of Riemann sum of Gaussian function

Hi this is my first question so far so I hope I'm doing it the right way. I'm trying to prove some result regarding the speed of congvergence of the Riemann sum $\Phi(R,\delta):=\sum_{k \in \mathbb{Z}...
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0answers
25 views

Riemann Sum of Ratio of Equation of Lines

Question: Can all kinds of summation be transformed into a Riemann Sum so that it can be transformed into a definite integral? Consider this limit: $$\begin{align} L=\lim\limits_{n\to\infty}\sum_{i=1}...
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0answers
72 views

Limit involving primes: $ \lim_{n\to\infty}\frac{1}{n}\sum_{k = 1}^{n}\left(\frac{1}{2}-\frac{1}{4}\frac{\log \log p_k }{\log p_k} \right)$

I have the following sum that I am trying to put into a cleaner formula (one that I can hopefully find the value of). This looks like it may be similar to a Riemann Sum, so it could turn into an ...
1
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0answers
54 views

A function with more than one number between the lower and upper riemann sums?

Given function $f$ that is continuous and defined on the closed, finite inverval $[a,b]$ Also given any two partitions $P_1$ and $P_2$ and their common refinement $P$ which consists of all of the ...