Questions tagged [riemann-sum]

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

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2answers
56 views

Another series involving $\log (3)$

I will show that $$\sum_{n = 0}^\infty \left (\frac{1}{6n + 1} + \frac{1}{6n + 3} + \frac{1}{6n + 5} - \frac{1}{2n + 1} \right ) = \frac{1}{2} \log (3).$$ My question is can this result be shown ...
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1answer
48 views

Is the definition of the Riemann sum from Thomas' Calculus correct?

I'm having trouble with theoretical understanding of the Riemann sum with this explanation/definition from Thomas' Calculus. I checked Wikipedia and it seems to state virtually the same.: On each ...
0
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1answer
34 views

A filter in $\mathbb{R}$ by means of the Riemann sums

I trying to find a filter $\mathcal{F}$ in $\mathbb{R}$ defined by means of the Riemann sums. In the other words, let $f: [0,1]\subset \mathbb{R} \longrightarrow \mathbb{R}$ be a continuous function. ...
7
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2answers
144 views

Is the sequence $x_n=\dfrac{1}{\sqrt{n}}\left(1+\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+\ldots+\dfrac{1}{\sqrt{n}}\right)$ monotone?

Observe that $x_1=1$ and $x_2=\dfrac{1}{\sqrt{2}}\left(1+\dfrac{1}{\sqrt{2}}\right)>\dfrac{1}{\sqrt{2}}\left(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{2}}\right)=1$. Thus, $x_2>x_1$. In general, we ...
0
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1answer
52 views

How near is the Riemann sum (without limit) to the definite integral

If $f$ is bounded and monotone in $[0,1]$ show that: $$\int_0^1 f(x)dx - \frac{1}{n}\sum_{k=1}^n f\bigg(\frac{k}{n}\bigg)=\mathcal{O}\bigg(\frac{1}{n}\bigg)$$ Well, by definition $$\int_0^1 f(x)dx=\...
1
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0answers
47 views

Is it true that $\lim_{n\to \infty}\sum_{i=1}^{n}\frac{1}{\sqrt{i}} - 2\sqrt{n} = 0$? [duplicate]

This problem originates from taking the limit of a Riemann Sum to evaluate an integral. In short, to help solve the limit, I used the bound $$\sum_{i=1}^{n}\frac{1}{\sqrt{i}} \leq 2\sqrt{n}$$ Which ...
4
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1answer
34 views

Defined integral as the limit of a Riemann sum is not always zero

I am first learning calculus and my mathematics notation lingo is not the best. I read that the defined integral could be written as the limit of a Riemann sum. Thus: $$ \int_{a}^{b} f(x) \,\mathrm dx ...
1
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1answer
37 views

How do we compute this sum using indefinite integrals?

Given that , $f\left( n \right)=\sum\nolimits_{k=1}^{n}{\ln {{\left( \frac{6n+2k}{n} \right)}^{\frac{2}{6n+2k}}}}$. How do we find the following limit $\underset{n\to \infty }{\mathop{\lim }}\,f\...
0
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1answer
42 views

summation for trapezoid

I had to compute $$T_2$$ using the trapezoid rule formula: $$ T_m (f) = \frac{h}{2}(f(a) + f(b)) + h \sum\limits_{i=1}^{2^m -1} f(x_i) $$ Values are the following: ...
1
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0answers
35 views

'Multiplicative' integration and 'Riemann products'

Warning! This question has little rigour and is entirely hand wavy crazy blue-sky speculative thinking so I apologise in advance. I was thinking about the gamma function and how it interpolates the ...
0
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1answer
30 views

Compute a limit based on a Riemann sum

From "Mathematical Analysis" of T.M. Apostol. \begin{gather*} \lim_{n \rightarrow \infty} \frac{b-a}n \sum_{k=1}^n f \left( a +k\frac{b-a}n \right) =\int_a^bf(x) \ dx \end{gather*} Use the ...
2
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0answers
34 views

Definite integral from Riemann Sum

I'm trying to convert this Riemann Sum into the definite integral, but I'm stuck. $$\lim_{n \rightarrow \infty } \sum_{k=n}^{2n} \sin\left(\frac{\pi}{k} \right)$$ I know the theory and I solved ...
1
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2answers
39 views

Evaluate $\lim_{n\to {\infty}} \sum_{j=1}^n \frac{j}{n^2 +j^2}$

The limit in question is $$\lim_{n\to {\infty}} \sum_{j=1}^n \frac{j}{n^2 +j^2}$$ and I'm trying to approach it via Riemann sums. I think a partition can be chosen ( although I'm not entirely sure ...
1
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1answer
26 views

Doubt about $\lim_{n \to \infty} \sum_{k=1}^{n} \left(1-\frac{1}{k+3}\right)$ and improper integral's Riemann sums

Evaluating the limit $$\lim_{n \to \infty} \sum_{k=1}^{n} \left(1-\frac{1}{k+3}\right)$$ I encountered a doubt: I did this $$\lim_{n \to \infty} \sum_{k=1}^{n} \left(1-\frac{1}{k+3}\right)=\lim_{n \to ...
2
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2answers
69 views

Find limit (Riemann sum)

$\displaystyle a_n=\sum^{n^2}_{k=1}\sqrt{\frac{k^2-k}{n^8+k^2n^4-kn^4}}$. I tried solving it by changing it a Riemann sum then integrating, however I couldn't manipulate the algebra to its form.
1
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1answer
20 views

Theorem needed to prove a summation

I know that the following relation holds: $$\sum_{x=1}^y\frac{x(5x+6)}{45}=\frac{y(y+1)(10y+23)}{270}$$ But what theorem should I use to prove that relation?
0
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1answer
14 views

When does the regular lower Riemann sum increases with n?

I would like to know, under which simple conditions on $f : [0,1] \longmapsto \mathbb{R}$ a continuous function, the Riemann regular lower sum, defined to be : $$ L_n(f) := \frac{1}{n}\sum_{k = 0}^{n-...
1
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0answers
20 views

Prove existence of subdivisions

Suppose that $f:[0:1] \rightarrow \mathbb{R} $ is a bounded function with upper and lower integrals: $$\int_{\underline{0}}^{1}f=0 \ \ \textrm{and } \int_{0}^{\overline{1}}f=1$$ (a) Prove that for ...
1
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2answers
82 views

Limit $\lim_{n→∞}\frac1n\sum_{j=c_n}^n\frac1{2n-2j+1}\sum_{k=1}^jk\prod_{i=k}^{j-1}\frac{2n-2i}{2n-2i+1}. $

If $c_n/n\to c>0$ as $n\to\infty$ (At the beginning of this question, I though the assumption is $c_n\to c$. With the help of metamorphy, the question will be ill-posed. So, the assumption has ...
0
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0answers
17 views

Prove zeta function has coninuous derivatives of all orders [duplicate]

I need to prove that the $\zeta$ function $$\zeta(x):=\sum^{\infty}_{k=1}\frac{1}{k^x}$$defines a function that has continuous derivatives of all orders. Here's my attempt: So, the $n$'th derivative ...
1
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1answer
62 views

Evaluate $n^2 \sum_{k=0}^{n-1} \sin \left(2\pi \frac{k}{n}\right)$ as $n \to \infty$

I must evaluate $$\lim_{n \to \infty} n^2 \sum_{k=0}^{n-1} \sin \left (2\pi \frac{k}{n}\right)$$ It reminds me a Riemann sum, so I'm trying to arrive to that $$\lim_{n \to \infty} n^2 \sum_{k=0}^{n-1} ...
0
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1answer
26 views

find $\lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{(k+1)^l}{k^{l+1}}, for: l\geq0$

find $\lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{(k+1)^l}{k^{l+1}}, for: l\geq0$ We mostly studied Riemann sums so my main idea is to somehow express $f(\frac{k}{n})$ for some f using this sum. I'...
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2answers
34 views

How do I evaluate this finite sum using simple techniques?

I am trying to calculate the definitive integral by definition (with Riemann sum). $$\int_{\frac{-\pi}{2}}^{\frac{3\pi}{2}} (2\sin{(2x+\frac{3\pi}{2})}) \ dx$$ But during the process of calculating ...
0
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0answers
24 views

Riemann sum approximation to Riemann integral

Let $f:[0,1]\to \mathbb{R}$ be a nonnegative Lipschitz continuous function. I know that $\frac{1}{T}\sum_{i=1}^T f(i/T)$ approximates $\int_0^1f(u)du$ as $T\to \infty$ and that $$\bigg\lvert \frac{1}{...
0
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1answer
43 views

Riemann Sum to integral form

Hi please help me in converting this Riemann Sum to integral form: Any help would be greatly appreciated..!! with $\Delta x={c\over an}$
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0answers
33 views

$\lim_{n\to\infty} ((1+1/n)(1+2/n)…(1+n/n))^{1/n}$ [duplicate]

Hi anyone knows how to evaluate this limit? I let $P$ = the limit Then $\log$ both sides to bring the $1/n$ power down. After that I integrate it by parts and gotten until $\log P = 2\log2 - 1$ ...
1
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1answer
44 views

Estimating integrals using Riemann sums

Let $f$ be a function such that $f''(x)<0$ for $x$ in $(0,1)$. We are required to find $$\left\lfloor\left(\frac{\sum_{r=1}^n \frac{f(r/n)}{n}+ \sum_{r=0}^{n-1} \frac{f(r/n)}{n}}{2 \int_{0}^{1} f(...
1
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3answers
67 views

How to prove that function $f$ is Riemann integrable

Let $f: [0,2] \rightarrow \mathbb{R}$ a bounded function with $$ f(x) = \begin{cases} x \qquad \qquad 0 \leq x \leq 1 \\ x-1 \qquad \quad 1 < x \leq 2 \end{cases} $$ Prove that $f$ is Riemann ...
1
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1answer
24 views

Integral Test: Anticipating That Integral is Lesser Than Riemann Sum

I am given the following $f(x)$ value: $f(x) = \frac{1}{x^2+x}$ The question says that it is possible to "anticipate" that the integral of $f(x)$ would be less than the sum of the series of $f(x)$. ...
0
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0answers
17 views

How to solve the following Riemann Integration problem

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f'$ is bounded. Given a closed and bounded interval $[a,b]$ and a partition $P=$ $\{a_0=a<a_1<........<a_n=b\}$ of $[a,...
1
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1answer
81 views

Limit of a sum of squared sines

Following up on this question Nice Limit $\lim_{n\to\infty}\sum_{k=1}^{n} \sin^2\left(\frac{\pi}{n+k}\right)$ . Fix a real $a$ and consider the sum: $$S(n)=\sum_{k=1}^n \sin^2\left(\frac{n^a\pi}{n+k}...
0
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0answers
12 views

Can a left-endpoint Riemann sum be described by a step function? How would you integrate it?

Can the left endpoint Riemann sum be described by a decreasing step function? I want to find the purple area in the graph from scratch using integration. How would I do it?
0
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1answer
34 views

Proof question about the definition of the integral via Darboux sums

I am having trouble on how to proceed with this question and would appreciate some help. $a,b,c: [f,g] \rightarrow \mathbb{R}$ are Riemann integrable, and $a(z) ≤ b(z) + c(z)$ for all z ∈ [f,g]. ...
0
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1answer
30 views

Computing the Darboux integral of $x$ on $[0,1]$ by definition.

I was wondering whether one can prove brute force without the notion of the Riemann sums the fact that $$ \underline{\int}_0^1xdx=\frac{1}{2} $$ I am trying to do this without any theorems about ...
1
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2answers
33 views

Given a continuous function $f$ prove $\lim\limits \frac 1n \sum_{k=0}^{n-1} k \int_{\tfrac kn}^{\tfrac{k+1}{n}} f(t)dt = \int_0^1 xf(x)dx$

Given a continuous function $f$ on $[0,1]$, it is asked to prove the follwoing $$\lim\limits \frac 1n \displaystyle\sum_{k=0}^{n-1} k \displaystyle\int_{\tfrac kn}^{\tfrac{k+1}{n}} f(t)dt = \...
3
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1answer
53 views

Calculate Upper Riemann Sum for $f(x)=x^{2}$ for $x \in [0,1]$

trying to find a way to calculate the upper sum $U(f,P)$ of $f(x)=x^{2}$ for $x \in [0,1]$ without the well known rules of integrals, just by partitioning the interval $[0,1]$ and performing a Riemann ...
0
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1answer
19 views

How to split pyramid volume into slices for Riemann sum.

I'm working on a problem: A pyramid of height H has a square base a by a. Find the area ...
-2
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1answer
34 views

Question from Riemann Integral [closed]

Evaluate: $$\lim_{h\rightarrow 0} \frac{1}{h} \int_{3}^{3+h}e^{{t}^2}dt$$ I tried this way
6
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1answer
178 views

$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{\infty} e^{-\frac{r^2}{2n^2}}$

Let $a_n = \frac{1}{n} \sum_{r=1}^{\infty} e^{-\frac{r^2}{2n^2}}$. The sequence is well-defined by considering the ratio test. What then is $\lim_{n \to \infty} a_n$? I suspect it is $\sqrt{\pi/2}$, ...
1
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1answer
76 views

Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal to $0$.

Let $f:[-1,1]\to\mathbb{R}$ be the function defined by $f(x)=\begin{cases} 1&\text{ if }x=0\\ 0&\text{ if }x\neq 0.\end{cases}$ Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal ...
0
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1answer
23 views

How to show that the upper sum of $x^2$ over [0,1] is always $\geq 1/3$ from first principles

In From Real to Complex Analysis, there is an exercise that is stated as follows: Using merely the definition of integrability, show that the function $f:[0,1]\to\mathbb{R}$ defined by $f(x)=x^2$ for ...
0
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2answers
33 views

How to multiply (simplify?) two or more Riemann sums?

$$\sum_{i=0}^{n-1}\ \sum_{j=i+1}^n\ \sum_{k=1}^j 1$$ I am studying The Algorithm Design Manual, so I have access to the process of simplification from http://www.algorist.com/algowiki/index.php/...
4
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1answer
81 views

Weird limit of a product $ \lim_{n\to \infty} \frac{3n+1}{3n}\frac{3n+2019+1}{3n+2019}…\frac{3n+2019n+1}{3n+2029n} $

I want to evaluate the following limit: $$ \lim_{n\to \infty} \frac{3n+1}{3n}\frac{3n+2019+1}{3n+2019}...\frac{3n+2019n+1}{3n+2029n} $$ The general term of the product is: $$ \frac{3n+2019k+1}{3n+...
-1
votes
1answer
50 views

Approximation of a sum for large $n$

May I know how I can approximate the following sum for large $n$: $$\frac{1}{n^2 xy}\left[\sum_{i=0}^{nx}\sum_{j=0}^{ny}\frac{1}{1 + \left(\frac{ |i-j|\wedge (n-|i-j|)}{n}\right)^2}\right]$$ where $...
0
votes
3answers
50 views

Using Darboux sums to find the value of an integral

I was given the following problem. Given $$ f:[0,1] \longrightarrow \mathbb{R}, f(x)= \begin{cases} 0 & 0\leq x\leq \frac{1}{2} \\ 1 & \frac{1}{2}< x \leq 1 \end{cases} $...
0
votes
0answers
36 views

prove that $\lim_{n\rightarrow \infty} S_NF(\frac{1}{\pi n})=\int_{0}^{1}\frac{\sin(\frac{t}{\pi})}{2t}dt$

I've been trying to prove that $\displaystyle\lim_{n\rightarrow \infty} S_NF(\frac{1}{\pi n})=\int_{0}^{1}\frac{\sin(\frac{t}{\pi})}{2t}dt$ this was in a quiz 2 years ago. I assume it somehow is ...
1
vote
2answers
63 views

How to explain this basic summation rule: $\sum_{i = 1}^n c = n\cdot c$, where $c$ is a constant? [closed]

The parallel rule for definite integrals , namely, $$\int_a^b c = c(b - a)$$ where $c$ s a constant, is rather intuitive, due to the fact that this number is computed in a way similar to the area of ...
2
votes
1answer
49 views

Difficulties with a limit while trying to calculate “ integral from 0 to 1 of x² dx”.

This is from an exercise in Stewart, Calculus. I managed to express the definite integral as the limit of a Riemann sum. After having calculated $\Delta x = \frac 1 n$ and right hand sample ...
1
vote
1answer
129 views

How to construct a Riemann integral from a Riemann sum that includes $(\Delta x)^2$

Suppose, for argument's sake, that we have a function $f: \mathbb{R}^2 \to \mathbb{R}$ that we want to integrate over a circular surface. We can use polar coordinates and cut up the $r$ and $\theta$ ...
-2
votes
3answers
60 views

Evaluate $ \lim_{n\rightarrow \infty} \sum_{r= 0}^{n} \frac{r}{n^2+ r} $

$$ \lim_{n\rightarrow \infty} \sum_{r= 0}^{n} \frac{r}{n^2+ r} $$ My attempt Divide Nr and Dr by $n^2$ $$ \lim_{n \rightarrow \infty} \sum_{r= 0}^{n} \frac{r/n^2}{1+ r/n^2} $$ =0 Is it ...

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