Questions tagged [riemann-sum]
This tag is for questions about Riemann sums and Darboux sums.
1,415
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Infimum of the Upper Sums
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, ...
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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 ...
- 471
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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 $$\Delta x = \...
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1
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Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable
Suppose $f : [0,1] \to \mathbb R$ be such that $$\displaystyle\sum_{i=1}^n |f(t_i) - f(t_{i-1})|^2 < 100$$ for any $n \in \mathbb N$ and $0 \leq t_0 < \cdots < t_n \leq 1$. Prove that $f$ is ...
- 3,237
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2
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Improper integral convergence implies the existence of an infinite series which its partial sum converges
Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a Riemann-integrable function at $[0, \beta]$ for each $\beta \in (0, \infty)$.
Suppose that $\forall x \in [0, \infty): \space f(x) \geq 0$.
If the ...
- 919
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Upper and Lower Sums for Negative Functions
The textbook I'm using to study integral calculus usually assumes for it's proofs that the function takes on only positive values. The author says that if we divide the x-axis into intervals, and pick ...
- 1,377
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1
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Convergence of Riemann sums with partitions with equidistant points
I am reading Spivak's book "Calculus".
The definition of integrable given there is the following:
$L(f,P)$ and $U(f,P)$ are the lower sum and the upper sum of $f$ for partition $P$.
I ...
- 2,416
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Does the upper bound of lower sums and the lower bound of higher sums depends on the type of intervals (open, closed or semi-open)?
I have consulted many books and notes on internet on the Riemann integral but in none of them have I found whether the upper bound of lower sums and the lower bound of higher sums depends on the type ...
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Why can't we compute this series using Riemann's sum?
Consider a series,
$$S=\sum_{n\ge0}\frac{(-1)^n}{2n+1}$$
Which can also be written as,
$$S=\lim_{n\rightarrow\infty}\sum_{r=0}^{n} \frac{2n}{(4r+1)(4r+3)}\cdot \frac{1}{n}$$
Substituting
$$\frac{r}{n}=...
- 467
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left and right Riemann sum comparison for $\int_2^\infty 1/x^2 dx$
I am having a confusion with left and right Riemann sums for the integral $\int_2^\infty 1/x^2 dx$.
We can compare this with the sum $\sum_{n=2}^\infty 1/ n^2 $. If we take the left Riemann sum then ...
- 5,033
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solution verification of $\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$
I asked this question
and I tried my own method which I am not sure if it is correct or wrong.
let $L=\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$
$$\ln(L)=\lim_{n\to\infty}\left(...
- 956
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Almost everywhere differentiable functions: what can we do? Integration by parts?
Let $A$ be an open measurable set and $A_0$ is a subset of $A$ which has measure zero.
Let $g$ and $f$ be almost everywhere differentiable function defined on $A$.
What is the impact of "almost ...
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Orthogonal functions and Riemann sums
Define inner product as $\langle f, g \rangle_{[a,b]} := \int_a^b f(x)g(x) \ dx$.
Say $f,g$ are orthogonal: $\langle f, g \rangle_{[a,b]} = 0 \Leftrightarrow \int_a^b f(x)g(x) \ dx = 0 \Leftrightarrow ...
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Difference Between Riemann Integrals and Definite Integrals
I'm currently using a textbook that covers Calculus and Analytic Geometry, however, the author seems to have confused me with notations and definitions. He first proves that the area under the curve ...
- 1,377
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Number of permutations for a conditional convergence series
I'm asking the next thing:
If we have a rational conditionally convergent series: $$\beta=\sum_{i=0}^\infty q_i \; \; \; ({q_i}\in\Bbb Q)$$
Then we know thanks to Riemann that it may be rearranged to ...
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Riemann Approximation
I have a question regarding the Riemann approximation. Suppose $a_{n,i}:=a_i=i/n$ is a sequence such that $$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^na_i=\int_0^1x\text{d} x.$$
The first ...
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Prove the following definite integrals as limits of sum $\int_a^b \frac{1}{x^2}dx = 1/a - 1/b$ [duplicate]
My attempt so far ..
$\int_a^b f(x)dx = \lim \sum_{1}^{n} h f(a+rh) $, where n tends to infinity, $nh = b-a$
Here $f(x) = \frac{1}{x^2}$ , using this we get
$\int_a^b f(x)dx = \lim \sum_{r=1}^{n^2} h ...
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Period function/Integrability | Real Analysis
Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ s.t $\forall a, \space b \in \mathbb{R}$ with $ a < b $, the function $f$ is Riemann
integrable at $[a, b]$ and satisfies: $ \forall \gamma \in \...
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Evaluate the limit $\lim_{n\to \infty }\sum_{i=1}^n \frac{1}{n} \cdot \lfloor \sqrt {\frac{4i}{n}} \rfloor$
I solved the problem using the Riemann integral. However, my answer did not match with the result given in the book. My answer was $\frac{3}{4}$ and the answer given in the book was just 3.
Help me ...
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Are upward directed set equivalent to downward directed set ? Example with $\supseteq$ and its dual relation $\subseteq$
If the set of partitions of $[a,b]$ $S$ is taken as the directed set, along with the order $\subseteq$, then the directed set has the property that "for every pair $(c,d)$ in the directed set ...
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How to convert $\lim_{n\to \infty} \sum_{k=1}^{n} \frac{1}{k + n}$ into a definite integral?
I am preparing for my Calculus exam, but am stuck on a question where I need to convert the limit of a Riemann sum to a definite integral.
I usually am able to tackle similar questions, but am stuck ...
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How to calculate $\int_0^1e^xdx$ using Riemann sums.
I need to calculate $\int_0^1 e^x dx$
using Riemann sums. And I'm getting stuck at the point where my intervals are $1/n$ and my final limit is
$\lim_{n \to \infty} \frac1n (1/(1-e^{1/n}))$ and at ...
3
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Regularizing infinite sum over $\sqrt{n^2+a^2}$
I am aware that one can use zeta function regularization to obtain
\begin{equation}
\sum_{n\in \mathbb{N}}n = -\frac{1}{12}
\end{equation}
I am wondering if it is possible to regularize a similar sum, ...
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Clarification on proof of theorem 1, appendix to chapter 13 in Spivak's Calculus
I am reading the appendix to chapter 13 in Spivak's Calculus (third edition, Cambridge University Press) and am stuck on the proof of theorem 1.
Theorem 1: Suppose that $f$ is integrable on $[a,b]$. ...
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Limit of Infinite Sum using Riemann sum
I was trying to learn about finding the limit of an infinite series using Riemann sums and I derived the following conclusion using the basic Riemann definition of definite integration:
$$\int_{0}^{k}...
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If $f$ is continuous on $I$, prove $f$ is Darboux integrable on $I$.
We are given: Suppose f is a real-valued function defined on the closed and bounded interval $I = [a, b] \subset \Bbb R$. And on the previous problem we proved that for each $\varepsilon > 0$, if ...
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Calculating equation of a curve using (Riemann) sums
I understand that Riemann sums serve to approximate integrals. But can we use this approximation, and the fundamental theorem of calculus, to work backwards to estimate the original equation of the ...
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Find limit with sum using integration
I'm working on this problem:
Find the limit
$$
\lim_{n \to \infty} \sum_{k=5n}^{7n} \frac{n}{k^2+n^2}
$$
My initial thought is to turn it into an integral and work from there, but I'm not sure how ...
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Explaining the conditional statement of Riemann Integral definition
A function $f:[a,b]\to\Bbb R$ is said to be Riemann integrable on $[a,b]$ if there exists a number $L\in\Bbb R$ such that for every $\varepsilon>0$ there exists $\delta_{\varepsilon}>0$ such ...
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Different definitions for Riemann integral
For an interval $[a,b]$, let ${\mathscr P}(a,b)$ be the set of all partitions $P=\lbrace x_0,x_1,\ldots,x_n\rbrace$ of $[a,b]$, where $a=x_0<x_1<\cdots <x_n=b$.
For a bounded function $f:[a,b]...
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Calculate limit of a sum as an integral [closed]
I need to calculate this limit as a definite integral but it doesn't look like Riemann sum at all:
$$ \lim_{x\to\infty} n^2 \sum_{i=1}^{n} \frac{1}{(n + i + 1)^3} $$
What would be an approach to this?...
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Proving that an integral is equal to a $\sum$
Let $f:[a,b]\to\mathbb{R}$ be continuous and bounded. I need to prove that for every $n\in\mathbb{N}$ there exist a sequence $\{x_i\}_{i=1}^n\subset[0,\pi]$ such that,
$$
\int_{0}^{\pi}f(x)\cdot n\...
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How to integrate $\sinh(x)$ as Riemann sum
I have a task where I need to calculate $\displaystyle \int_{0}^{1} \sinh(x) \,dx$. I know that to integrate $\displaystyle \int_{0}^{\pi/2} \sin(x) \,dx$ we take $\Delta x_i = \dfrac{\pi }{ 2n}$ and $...
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How to prove that a sum is a Riemmanian sum
Lets say that we have the well known Gauss-Chebyshev Quadrature rule of the first kind i.e,
$\int^1_{-1}f(x)\frac{1}{\sqrt{1-x^2}}dx= \sum^n_{k=1} w_kf(x_k) +R_n(f) $, where the nodes $x_k$ are $x_k=\...
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Evaluation of a Riemann sum to calculate the integral of $x\ln(1+x)$ from $0$ to $1$
I have encountered a problem in which you are asked to calculate the integral $\int_0^1x\ln(1+x)$ using a Riemann sum. The Riemann sum is $\sum_{k=1}^n\frac{k}{n^2}\ln(1+\frac{k}{n})$, and it should ...
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Evaluating the limit of two sums/sequences.
Let me first state the question and then provide some context and my approach. I'm trying to evaluate limit of the two following sequences :
$$
S_n = \sum_{1\leq k \leq n}\frac{2k^2-3kn-3n^2}{(k+3n)(...
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Fake proof that all functions are integrable [duplicate]
Below I present a proof that I know is wrong. It "states" that all bounded functions are integrable. However, I am not sure why it is wrong, i.e., I am unsure where my logic fails. It goes ...
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Proof Check: Equivalence of Riemann and Darboux Integrals
I'm practicing my $\delta$-$\epsilon$ proofs by verifying that the Riemann and Darboux integrals are equivalent for functions on a closed interval. Here are the definitions I'm working with (taken ...
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Riemann Sum of $\int_1^b x^{-1/2}\,dx$ with $x^*_i=\frac{(\sqrt{x_{i-1}}+\sqrt{x_i})^2}{4}$
I am trying to evaluate the integral
$$\int_1^b x^{-1/2}\,dx$$
with $b\in\mathbb{Z}_{>1}$ using Riemann sums. I consider the uniform partition
$$\mathcal{P}_n=\left\{1,1+\frac{b}{n},1+\frac{2b}{n},\...
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Proving a piecewise function is not Riemann integrable
$f: [0,1] \to \mathbb{R}$ where
\begin{equation}
f(x)=
\begin{cases}
4x^3 & \text{if } x \in \mathbb{Q}\\
0 & \text{if } x \in \mathbb{R}\setminus\mathbb{Q}
\end{cases}
...
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Justification for a substituion that turns a finite sum to infinite - constructing the Grunwald-Letnikov fractional derivative (Fractional Calculus)
Steps in question
These steps raise numerous questions.
What is the reasoning behind choosing $\delta _Nx\equiv [x-a]/N$ ?
This seems almost arbitrary. I understand that $a$ and $x$ eventually ...
- 1
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Du Bois-Reymond criterion for Riemann-integrability
Function f \in \textbf{R} ([a,b]) (1) \Leftrightarrow f is bounded on [a,b] and for all ...
1
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29
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Transforming Riemann Sum into Integral Form
I am a bit struggling with transforming a Riemann's sum into its definite integral form.
I am given the following equation:
$$
\sum_{i=1}^n\left(\left(3-\frac{i}{n}\right)^3\right) \frac{1}{n}
$$
Now, ...
1
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1
answer
51
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A Riemann Sum clarification
I have to find the area under the curve $f(x) = x^2$ in the segment $[-2, 1]$ using Riemann Sum. So here is what I did, but it's wrong and I need some clarification about (see the questions in the end)...
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How to show that a sum is inclosed by two values / bounds?
Show that this holds for all c>0
$\frac{\pi}{2\sqrt{c}} \le \sum_{n=0}^{\infty} \frac{1}{n^2 + c} \le \frac{\pi}{2\sqrt{c}} + \frac{1}{c}$
I'd really appreciate it if some can tell me if my proof ...
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How can i compute this limit related with Riemann sums?
How can i compute this limit: $\lim_{n \to +\infty} \sum_{k=1}^{n} f(\frac{k-1}{n})(f(\frac{k-1}{n})-f(b_k ))$, where f:[0,1]->R, f differentiable function with continuous derivative and $\int_{\...
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Prove that $\int_a^b$ $ \int_c^d $ $f(x)$ $g(y)$ $ dydx$ $= $ $(\int_a^b$ $f(x)$ $ dx)$ $(\int_c^d $ $g(y)$ $ dy)$ solution verification
Using the techniques that I have been taught, I have attempted to prove the following result:
$$\int_a^b \int_c^d f(x)g(y) \space dydx = \left(\int_a^b f(x) dx\right)\left(\int_c^d g(y) dy\right)$$...
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Summation bound above integral
How is this true?
$\frac{\sqrt{k}}{k^2}+\frac{\sqrt{k+1}}{(k+1)^2}+\cdots+\frac{\sqrt{n}}{n^2} \leqslant \int_{k-\frac{1}{2}}^{\infty} \frac{\sqrt{x}}{x^2} d x$
where $k,n$ are natural numbers. I can ...
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Proving that the image of a Riemann sums of a function with the intermediate value property(Darboux function) is an interval
Let $a$ and $b$ be real numbers, such that $a < b$. We say $\Delta$ is a division of the interval $\left[a, b\right]$, if $\Delta=(x_0, x_1, x_2, ..., x_n)$, for some non-zero natural number $n$, ...
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From Riemann Sum to Definite Integral
I have the sum
\begin{align*}
\sum_{j=1}^T\frac{1}{h}g\left(\frac{\omega_j-\omega}{h}\right)f(\omega_j),\quad \omega_j = \frac{2\pi j}{T}
\end{align*}
where $h>0$ and $\omega \in[0,\pi]$. I'm asked ...
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