All Questions
Tagged with riemann-sum solution-verification
48 questions
1
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0
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42
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Spivak calculus and a Riemann sum of a product of functions
I am trying to understand Spivak's development of integration in Calculus, 4th ed. The integral is introduced in Chapter 13, and I find the proofs quite difficult and unfamiliar. The Appendix to the ...
- 392
0
votes
1
answer
50
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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 ...
- 528
1
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0
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59
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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 $L=\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$
$$\ln(L)=\lim_{n\to\infty}\left(...
- 6,781
3
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1
answer
245
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Prove that $\int_a^b$ $ \int_c^d $ $f(x)$ $g(y)$ $ dydx$ $= $ $(\int_a^b$ $f(x)$ $ dx)$ $(\int_c^d $ $g(y)$ $ dy)$
Using the techniques that I have been taught, I have attempted to prove the following result:
$$\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)$$...
- 293
3
votes
1
answer
73
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(Verifying) Riemann Integral of $g(x)=x$ on $[0,1]$ using tagged partitions.
I need to verify whether my method is correct or not in the following problem.
Let $g(x)=x$ on $I=[0,1]$.
Let
$
\dot{\mathcal{P}} = \left\{ \left( \left[ x_{i-1},x_i \right],t_i \right) \right\}_{i=...
1
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2
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127
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Showing that surface area is equivalent to $\int_{S}\|\partial_u\phi\times\partial_v\phi\|dudv$, and is there MVT for bijections: $\Bbb R\to\Bbb R^2$?
$\newcommand{\d}{\,\mathrm{d}}$It can be shown that arclength, considered as a sum of increasingly fine partitions of the graph, approaches the integral formulation. However, I have only ever seen the ...
- 44.7k
3
votes
1
answer
98
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Integral-sum conversion in the proof that the $\chi^2$ statistic tends to the $\chi^2$ distribution
I cite the derivation on Wikipedia, here. The focus of that section of the article is to show that the $\chi_p^2$ statistic is asymptotically equivalent to the $\chi^2$ distribution.
Let $n$ be the ...
- 44.7k
1
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0
answers
144
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Show Riemann-integrability of discontinuous function
Let's assume that $f:[a,b]\to\mathbb{R}$ is bounded, i.e. $\Vert f\Vert_{\infty}\leq M$ and that is has countable infinitely many points where it is discontinuous.
Show that $f$ is Riemann-integrable ...
- 4,799
1
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1
answer
49
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$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2}$ the problem of $k = 0$. Am I doing it right?
I would like to calculate:
$$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2}$$
We have that:
$$\lim_{n \to \infty} \sum_{k=0}^n \frac{\sqrt{2n^2+kn-k^2}}{n^2} = \lim_{n \to \infty} \...
- 287
1
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1
answer
69
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$ \lim_{n \to \infty} \sum_{k=1}^n \sqrt{2- \left( \frac{k}{n} \right)^2} \cdot \frac{k}{n^2}$ is my solution any good?
First write
$$ \lim_{n \to \infty} \sum_{k=1}^n \sqrt{2- \left( \frac{k}{n} \right)^2} \cdot \frac{k}{n^2} = \lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^n \sqrt{2- \left( \frac{k}{n} \right)^2} \cdot \...
- 287
4
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1
answer
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(Proof-check) for the expression of total variation of a continuously differentiable function
There are many questions on the same topic like in this thread; however, the part where I am stuck is not explained anywhere. This problem is from the book Understanding Analysisby Stephen Abbott.
...
4
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2
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275
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Using definite integral to find the limit.
I would like someone to verify this exercise for me. Please.
Find the following limit:
$\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\right)$
$=\lim\limits_{n \to \...
- 1,161
2
votes
1
answer
42
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Partition of sum of two bounded functions
Following my previous question about Riemann-Stieltjes integration, I'm asking this problem.
Let $f_1$ and $f_2$ be bounded on $[a,b]$ and $\alpha$ is increasing on $[a,b]$. Define $f = f_1 +f_2$. ...
- 4,402
1
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1
answer
46
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Partition and Riemann-Stieltjes integration
Let $f$ and $g$ be bounded on $[a , b]$ and $g \in \mathcal{R}(\alpha)$ on $[a ,b]$. Also $P$ is an arbitrary partition. If $f\le g$ and $\alpha$ is increasing on $[a , b]$ determine whether the ...
- 4,402
2
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1
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158
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Let $f\in C([a,b])$, $\int_{a}^{b}f(x)\phi'(x) \ dx=0$ for all continuous and differentiable functions $\phi$. Show $f$ is constant
Let $f\in C([a,b])$ and $\int_{a}^{b}f(x)\phi'(x) \ dx=0$ for all continuous and differentiable functions $\phi$ on $[a,b]$ with $\phi(a)=\phi(b)=0$. Show that $f$ is constant on $[a,b]$.
I would ...
- 1,697
0
votes
4
answers
149
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Proof that $\sum_{k=1}^n \frac{k}{2^k} < 2 $ for all $n \geq 1$
I recently completed a variation of a problem I found from a mathematical olympiad which is as follows:
Prove that, for all $n \in \mathbb{Z}^+$, $n \geq 1$, $$\sum_{k=1}^n \frac{k}{2^k} < 2 $$
I ...
- 117
0
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0
answers
23
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Proving the Montonicity Properties of Lower and Upper Sums
Here's what I'm trying to prove.
Let $f: [a,b] \to \mathbb{R}$ be a bounded function. Let $P$ and $Q$ be partitions of $[a,b]$ such that $P \subseteq Q$ ($Q$ is a refinement of $P$). Then:
$$L(f,P) \...
- 6,378
0
votes
1
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142
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Finding an integral by evaluating a Riemann Sum
Below is a problem I did. I believe I got the right answer. However, I am not convinced my method is correct. I am hoping somebody here can verify that my solution is correct or tell me where I am ...
- 4,168
9
votes
3
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311
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Show that $\lim\limits_{N\rightarrow\infty}\sum\limits_{n=1}^N\frac{1}{N+n}=\int\limits_1^2 \frac{dx}{x}=\ln(2)$
Show that:
$$\lim\limits_{N\rightarrow\infty}\sum\limits_{n=1}^N\frac{1}{N+n}=\int\limits_1^2 \frac{dx}{x}=\ln(2)$$
My attempt:
We build a Riemann sum with:
$1=x_0<x_1<...<x_{N-1}<x_N=2$
$...
- 1,011
1
vote
2
answers
177
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The closed form solution to the equation with sum of exponential functions
I have a function
$\begin{equation}f(x)=\sum_{k=1}^{n}\left(\frac{1}{n}+\frac{ax}{k}\right)e^{-a(\frac{nx}{k}-b)}\end{equation}$,
where $a,b$ are both positive constants, $n$ is a positive integer.
$x^...
- 11
1
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0
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26
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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 ...
- 33
1
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1
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93
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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:
$$S(n)=\sum_{k=1}^n \sin^2\left(\frac{n^a\pi}{n+k}...
- 4,069
0
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1
answer
40
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Proof question about the definition of the integral via Darboux sums
I am having trouble on how to proceed with this question and would appreciate some help.
$a,b,c: [f,g] \rightarrow \mathbb{R}$ are Riemann integrable, and $a(z) ≤ b(z) + c(z)$ for all z ∈ [f,g]. ...
user763922
1
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1
answer
98
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Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal to $0$.
Let $f:[-1,1]\to\mathbb{R}$ be the function defined by
$f(x)=\begin{cases}
1&\text{ if }x=0\\
0&\text{ if }x\neq 0.\end{cases}$
Prove that the upper integral $\int_{-1}^1f(x)dx$ is equal ...
- 349
2
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0
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329
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Show that $\int_a^b cf = c \int_a^b f$
Let be $f$ and $cf$ integrable functions on the interval $[a,b ]$ where $c$ is a constant with $c<0$.
Show that: $$\int_a^b cf = c \int_a^b f.$$ I know that there is a different approach where ...
- 4,799
0
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0
answers
65
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Integrability criteria: $U(f, P) -L(f, P)<\epsilon$
In our lecture we have defined integrability as follows
Let $f:[a, b] \to \mathbb{R}$. We say that $f$ is integrable if
$sup\{L(f, P), P$ is a partitition of $[a, b]\}= inf\{U(f, P), P$ is a ...
- 4,799
2
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1
answer
32
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Verification of a basic Riemann summation problem: $f(x) = 1+x$ where $x\in [-1, 2]$
This is the very first time I solve a problem involving Riemann's summation so I would like to verify whether I get it correctly. Problem statement (hopefully, I have translated the problem statement ...
- 5,437
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1
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30
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Trying to approximate Riemann sum with $\sin(\frac{x}{5})$
Let $P= 0, \frac{1}{2},1,2$
Find the upper, lower, and exact area and whether the lower or upper sum is more accurate
Starting with the left:
LH: $$\sin(0)(\frac{1}{2}-0)+\sin(\frac{.5}{5})(1-\frac{...
- 2,411
1
vote
1
answer
39
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Dissection and Integrals
Let $[a,b]$ be an interval. Let $D_1,D_2$ be arbitrary dissections of $[a,b]$. Then $U(f,D_1 \cup D_2)$ $=$ $U(f,D_1)+U(f,D_2)$.
My proof so far:
Suppose $D_1$ $=$ $\{$ $a,x_1,x_2.....,x_n=b$ $\}$ ...
user643073
1
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0
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28
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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 ...
- 12.9k
0
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0
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58
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Sum inf of a partition is less or equal than sum inf of a finer partition
Let $I=[a_1,b_1]\times...\times[a_n,b_n]$ an interval in $\mathbb{R^n}$ and $P=\{P_1,...,P_n\}$ a partition of $I$. Let $f:I \rightarrow \mathbb{R}$ be a bounded function in $I$. For each $J \in P$ we ...
- 2,525
2
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1
answer
68
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Riemann Sum. Check proof.
Let $f:[a,b] \longrightarrow \mathbb{R}$ and $\displaystyle\sum(f; P^{\ast})$ a Riemann sum. Prove that, if $\displaystyle \lim_{|P|\to 0}\sum(f; P^{\ast}) = L$, then $f$ is a limited function
$\...
- 4,125
1
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0
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118
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Proving $f$ is Riemann integrable iff for every $\varepsilon >0$ and partitions $P$ and $S, U(f,P) - L(f,S) < \varepsilon$
Now I see this question as naturally applying Riemann's condition, which states that $f$ is Riemann integrable iff $U(f,P) - L(f,P) < \varepsilon$. I think this kind of automatically solves itself ...
- 123
0
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1
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69
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Riemann Stieltjes sums / norms
Let $f \in \Re[0,1]$. Prove that $\lim_{n \to \infty} \sum_{k=1}^n f(\frac{k}{n}) \frac{1}{n} = \int_0^1f.$
I want to prove this using Riemann-Stieltjes sums. Here is what I thought so far:
I know ...
- 123
0
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1
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402
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Explanation of proof about inequality of Darboux-Sum?
I'm confused with the terminology from my book:
LEMMA
If $P$ and $Q$ are partitions of $[a,b]$, then $L_f(P)\le U_f(Q)$.
(proof omitted)
From this lemma it follows that the set of all lower ...
2
votes
2
answers
439
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Prove if f is positive and increasing on $[a, b]$ then $L_n ≤ A ≤ R_n.$ (riemann sum)
Prove if f is positive and increasing on $[a, b]$ then for all $n\ge 0$ we have $L_n \le A \le R_n$. (Riemann sum)
Let $A$ denote the actual area.
Let $L_n$ denote the left Riemann sum.
Let $R_n$ ...
- 6,605
2
votes
1
answer
374
views
Show $\lim\limits_{n\to\infty}\left[\frac{1}{n}\sum_{i=1}^{n}f\left(\frac{i}{n}\right)\right]=\int_{0}^{1}f(x)\,{\rm d}x$
Suppose that the function $f:[0,1]\rightarrow\mathbb{R}$ is integrable. Prove that $$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\left[f\left(\frac{1}{n}\right)+f\left(\frac{2}{n}\right)+\cdots+f\left(\...
- 3,651
1
vote
0
answers
254
views
Prove Darboux integrable
Prove that if $g(x):=0$ for $0\le x\le1/2$and $g(x):=1$ for $1/2\lt x\le1$ then the Darboux Integral of $g$ on $[0,1]$ is equal to $1/2$.
My answer: Lets define a partition $P_\epsilon =(0,1/2,1/2+\...
- 59
1
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1
answer
1k
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Alternative proof of the Riemann Sum Theorem using Mean Value Theorem for Integrals.
I've been reviewing proofs for a couple of calculus theorems and as I was trying to recall the proof of the Riemann Sum Theorem which uses Lower Sums and Upper Sums I came up with an idea to prove it ...
3
votes
1
answer
277
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Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$, where $f(x) = 1+x^2$
Given $f(x) = 1+x^2, \alpha(x) = x^3, x \in [-1,1], P = \{-1,\frac{-1}{2},0,\frac{1}{2},1\}$. Find $U(P,f,\alpha)$ and $L(P,f,\alpha)$.
Ok, for this problem, we have $\alpha (x) = x^3$. And, I am ...
- 1,703
1
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0
answers
103
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To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$ ( Riemann Integration)
To prove $| S(f,P,T) - S(g,P,T) | \leq M(b-a)$
Question : Let $[a,b] \subseteq R$ be a non degenerative closed bounded interval and let $f,g :[a,b] \rightarrow R$ be functions .Suppose that there is ...
- 3,468
0
votes
0
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746
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Find the upper sum and lower sum for the following function with respect to the given partition
I have the following homework problem:
Find the upper sum and lower sum for the following function with respect to the given partition:
Let $s:[0,1]\rightarrow \Bbb R$ be defined by: $$s(x)=\bigg\...
- 399
0
votes
1
answer
959
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Riemann Integral Property for Continuous, Monotonic, Non-negative Function
If $f$ is continuous, non-negative, and monotonically increasing function on $[0,∞)$, then prove that $\int^{x}_{0} f(t)dt\leq xf(x)$ $\forall x ≥ 0$
My attempt:
Define $F(x)=\int^{x}_{0} f(t)dt$. ...
- 1,127
5
votes
2
answers
2k
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Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$.
Exercise:
Suppose that $a<b$ and that $f:[a,b]\rightarrow R$ is continuous. Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$.
attempt of ...
- 2,223
2
votes
1
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152
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Let $S_n:= \frac{b-a}{n}\sum_{i=1}^{n}f(t_{i,n})$. Prove: $\lim_{n\to\infty}S_n = \int_a^bf(x)\ dx$.
I will post the assignment and then my attempt at solving it.
Let $a,b \in \mathbb{R}$ with $a<b$ and let $f: [a,b] \rightarrow \mathbb{R}$ be a continous function. We'll now define a sequence ...
- 1,354
3
votes
0
answers
54
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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{\...
1
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0
answers
151
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Non-negative, continuous function with integral [duplicate]
Let there be an integrable, non-negative function $f$ in a range $[a,b]$.
If the integral $\int_a^b f(x) \, dx$ equals $0$, prove that $f(x)=0$ for every $x$ for which $f$ is continuous.
I have ...
- 93
8
votes
1
answer
295
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Question on Riemann sums
Question is :
What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$
So, at first sight i am afraid this is not even related to Riemann integration of $f$ ...
user87543