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Questions tagged [riemann-sum]

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

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3answers
69 views

Limit of $x_n=\sum_{k=np+1}^{nq}\frac{1}{k}$ using Riemann sum

I am trying to find the limit of the following sequence using Riemann sum: $$x_n=\sum_{k=np+1}^{nq}\frac{1}{k}\qquad p,q\in\mathbb{N}\quad p<q$$ I have tried to develope the expression: $$\frac{1}{...
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0answers
14 views

Maximizing the Riemann sum for partitions of fixed size

As I am doing again some elementary maths (for teaching), I have this following problem regarding Riemann sums. Let's say we consider a function $\,f$ on $[0,1]$ and we only consider partitions of ...
2
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2answers
36 views

Partitions and Riemann sums

Reading my textbook and i'm alittle bewildered by a step in calculating the Riemann sum. The question reads as follows: "Calculate the lower and upper Riemann sums for the function $f(x)= x^2$ on ...
2
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2answers
59 views

Limit of sum as definite integral

I don't understand why $$\displaystyle \sum_{k=1}^n \dfrac{n}{n^2+kn+k^2} < \lim_{n\to \infty}\sum_{k=1}^n \dfrac{n}{n^2+kn+k^2}$$ whereas $$\displaystyle \sum_{k=0}^{n-1} \dfrac{n}{n^2+kn+k^2} &...
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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 ...
1
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1answer
28 views

Evaluate the limit for a function defined on [0,1]

The limit is a Riemann sum $$\lim_{n\rightarrow\infty}\sum_{i=1}^n\left(\frac{i^4}{n^5}+\frac{i}{n^2} \right)$$ $\delta x=\frac{1}{n}$, so I distribute it to the terms to get $$\lim_{n\rightarrow\...
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1answer
42 views

Consider the Integral $ \int_{0}^1\left( x^3-3x^2\right)dx $ and evaluate using Riemann Sum

Consider the integral $$\int_{0}^1\left(x^3-3x^2\right)dx$$ $\delta x=\frac{1}{n}$ $x_i=0+\frac{1}{n}i$ Plugging everything in I get $$\lim_{n\rightarrow\infty}\sum_{i=1}^n \left(\frac{1}{n}i \...
2
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1answer
64 views

Evaluate integral $\int_{-2}^0 x^2+x\ dx$ using Riemann Sum

Consider the integral $$\int_{-2}^0 x^2+x\ dx.$$ The question says to use Riemann Sum theorem which is $$\sum_{i=1}^nf(x_i)\delta x$$ I know that $\delta x= \frac{-2}{n}$ and that $x_i=-2+(\frac{2}{n}...
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0answers
65 views

Prove $f$ is integrable on $[a,b]$ if $f$ has finitely many accumulation points on $[a,b]$.

Suppose interval $I\in[a,b]$ as finitely many limit points, $f:[a,b]\to \mathbb{R} $ is bounded on $[a,b]$ and continuous on $[a,b]\setminus I$. Use the fact that if $f$ is continuous except at ...
2
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1answer
37 views

How to evaluate the sum for definite integrals using limit definition?

If $f$ is integrable on $[a,b]$, then $$ \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i) \Delta x_i $$ where $\Delta x = (b-a)/n$ and $x_i = a + i\Delta x$. Use this definition of the ...
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1answer
31 views

Proof area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$

I'm supposed to prove that the area of {$(x,y) R^2 | 0 \le y < e^x$ and $0\le x< h$} is $e^h-1$ I was going to try to make it a function and calculate it using a Riemanns sum. That led me to ...
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1answer
29 views

Riemann sums over dense countable sets

Let $f$ and $g$ be positive, smooth and integrable functions in $\mathbb{R}$, whose derivatives are also integrable. Assume as well that the expression $$ \frac{\sum_{q\in \mathbb{Q}} f(q)}{\sum_{q\in ...
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2answers
78 views

How to show $\lim_{n\to\infty}n\left\{\sum_{k=1}^n\frac{1}{(n+k)^2}\right\}=\frac{1}{2}$

Show that $$\lim_{n\to\infty}n\Bigg\{\dfrac{1}{(n+1)^2}+\dfrac{1}{(n+2)^2}+\dfrac{1}{(n+3)^2}+\cdots+\dfrac{1}{(2n)^2}\Bigg\}=\dfrac{1}{2}.$$ Proof: We can rewrite $$\lim_{n\to\infty}n\Bigg\{\...
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1answer
40 views

Show (F$_{k}$) converges uniformly to some continuous function

Suppose ${0<r<1}$. For each k $\in$ $\mathbb{N}$, define F$_{k}$ $\in$ C$\bigl($[-r,r]$\bigr)$ by F$_{k}$(x) = $\sum_{n=1}^k$ x$^{n}$. Show (F$_{k}$) converges uniformly to some continuous ...
3
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2answers
63 views

Limit, Riemann Sum, Integration, Natural logarithm

For any natural number $m$, $\lim_{n\rightarrow \infty }\left ( \frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots +\frac{1}{mn} \right )=\ln (m)$. I tried to prove the statement in the following way. ...
0
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1answer
40 views

How to show $\lim_{n\to\infty}\sum_{k=1}^n(1+(k)/n)^3(1/n)=15/4$?

How would I show $$\lim_{n\to\infty}\sum_{k=1}^n(1+(k)/n)^3(1/n)=15/4?$$ My attempt is using the Riemann Sum technique. We know $(1+(k)/n)^2=f(\zeta_k)$ and $(1/n)=\Delta x$. So the definite integral ...
12
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1answer
317 views

Has the Riemann Rearrangement Theorem ever helped in computation rather than just being a warning?

A decent course in elementary analysis will eventually discuss series, absolute convergence, conditional convergence, and the Riemann Rearrangement Theorem. However, in any presentation I've seen, in ...
1
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1answer
50 views

Prove that for any $\epsilon > 0, \exists \delta > 0,$ if $||P|| < \delta $, then $|L(f,P) - I|<\epsilon $ , and $|U(f,P) - I|<\epsilon $

Let function f be integrable on [a,b] and $I = \int_{a}^{b} f(x) dx.$ Then, for any $\epsilon > 0, \exists \delta > 0,$ such that if P is any partition of [a,b] and $||P|| < \delta $, then $|...
3
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1answer
89 views

Calculate $\int_{0}^{1} x^2 dx$ using the definition of the integral using Riemann Sums

Okay, so my Real Analysis textbook defines a definite integral as follows: Let $[a,b]$ be an interval and $f$ a function with domain $[a,b]$. We say that the Riemann sums of $f$ tend to a limit $l$ ...
2
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1answer
40 views

Product of the Riemann sum

Hi I want to prove that for all $x,y\in R$, the following holds $$(\sum_{n=0}^{\infty}\frac{x^n}{n!})(\sum_{n=0}^{\infty}\frac{y^n}{n!})=\sum_{n=0}^{\infty}\frac{(x+y)^n}{n!}$$ without using $e$ ...
0
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1answer
43 views

$\sum \exp(-x^2)$ vs $\sum x^2 \exp(-x^2)$

I am curious about the following sum, for $\alpha \in (0,1)$: $$\sum_{k = -\infty}^{\infty} (1-(2k - 1 + \alpha)^2) \exp(-\frac{1}{2} (2k - 1 + \alpha)^2)$$ I have reasons to believe sum should be ...
1
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0answers
27 views

If $0 \le f(x) \le M$, prove $\lim\limits_{n \to \infty} \left[ \int_0^1 f(t)^n \, dt \right]^{1/n} = M$ [duplicate]

Q: Suppose that $f$ is a continuous, nonnegative function on the interval $[0,1]$. Let $M$ be the maximum of $f$ on the interval. Prove that: \begin{align*} \lim\limits_{n \to \infty} \left[ \int_0^...
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0answers
33 views

Show the relationship between the supremum and infimum of f^2 and |f|

Suppose f: [a,b] $\to$ $\mathbb{R}$ and B satisfy |f(x)| $\le$ B for every x $\epsilon$ [a,b]. Show that if P = {x$_{0}$,...,x$_{n}$} is a partition of [a,b], then M(f$^{2}$,[x$_{i-1}$,x$_{i}$]) - ...
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0answers
53 views

Help for this problem involving rieman integral and partitions

If $f: I--->\mathbb R$ is bounded, let $||f||:= Sup {|f(x)|: x \in I}$, and if $P =(x_0,...,x_n)$ is a partition of $I:=[a,b]$, let $||P||:=Sup [x_1-x_0,...,x_n-x_{n-1}]$ (a) If $P'$ is the ...
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0answers
56 views

Prove the following are equivalent

Let $f$ be bounded on $[a,b]$ and integrable on $[c,b]$ for $a<c<b$. I need to prove the following are equivalent: a) $\lim\limits_{x\to a+}\int_{x}^{b}f$ exists in $\mathbb{R}$ b)$\lim\...
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}{...
3
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0answers
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 ...
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1answer
31 views

Calculate performance of deleting each first element from vector, calculate sum

My friend told me that deletion of all the elements from vector in big O notation is quadratic performance is worst case (big O notation). The worst case requires to delete always the first object, ...
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2answers
27 views

Limit of the sequence including log. [closed]

Define a sequence $s_n$ of real numbers by $s_n = \sum_{i=1}^n \frac{(\log(n+i)-\log n)^2}{n+i}$. Does $\lim_{n \to \infty} s_n$ exist? If it, compute the value of this limit. Any idea or hint?
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1answer
65 views

Show $s(f) = \int_{\_a}^b f $

Show $s(f) = \int_{\_ a}^b f $ Where $s(f) = sup \{ \int h d\lambda : h \in C[a,b], h \leq f\}$ And $\int_{\_ a}^b f $ is the Lower Darboux Integral edit I know that $f: [a,b] \rightarrow R$ is ...
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1answer
24 views

A question regarding two Cauchy definitions of integrability

While doing research on the construction of the Riemann integral I've stumbled upon two different criterions for Riemann integrability. In this article (on page $19$) there is a theorem called the ...
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2answers
52 views

How to prove $L(f, P_{1}) \leq U(f, P_{2})$ under given conditions.

I would like to show that under the assumptions of the following Theorem, if $P_{1}$ and $P_{2}$ are partitions of $[a, b]$ then $L(f, P_{1}) \leq U(f, P_{2})$, and I would like to use this result to ...
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0answers
30 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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1answer
57 views

Approximating Darboux integrals with continuous functions

Let $f:[a,b]\rightarrow\mathbb{R}$ be a bounded function. Show that the upper Darboux integral of $f$ is equal to the infimum of the Riemann integral of $g$ over all continuous functions $g\geq f$. ...
8
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2answers
270 views

Perfect understanding of Riemann Sums

I am not sure I have completely and properly understood Riemann sums. Given a sum like: $$S_n = \displaystyle\sum_{r=1}^n \dfrac{r^4+ r^3n +r^2n^2 +2n^4}{n^5}$$ After dividing by $n^4$ we will ...
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1answer
53 views

A property of a Riemann Integrable function

It was posted in an answer to my previous question that If a function $f$ is Riemann-integrable on $[0,\infty)$, then, $$\int_0^\infty f(x)\,dx = \lim_{h\rightarrow 0}\,h\sum_{n=0}^\...
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2answers
451 views

Evaluate $ \lim_{h \to 0} h\sum_{n=0}^{\infty} e^{-n^2h^2}$ [closed]

Evaluate $$ \lim_{h \to 0} h\sum_{n=0}^{\infty} e^{-n^2h^2}$$ I think it is somehow related to Riemann Sums, but I'm not sure. Please help.
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2answers
33 views

a function $=x(1-x)$ on rationals, and $=0$ on irrationals is not Riemann integrable on $[0,1]$

Let $f(x)=x(1-x), x\in [0,1]\cap \Bbb Q$, and $=0,\ x\in [0,1]\backslash \Bbb Q$. Can we show that $f(x)$ in not Riemann integrable on $[0,1]$. If the inteval is $[a,b]$ with $0<a<b<1$, ...
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0answers
54 views

I'ts true that this sum is equal to this integral?

I need to know how that sum $$\sum_{k=1}^n \frac{1}{(1+i)^{T_k}}*({T_k}-{T_{k-1}})$$ Can be equal to the integal $\displaystyle \int^{{T}}_{0} {\frac{1}{(1+i)^{s}}}ds$ Where $T_0=0$ and $T_n=T$ and ...
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0answers
22 views

Upper and lower Riemann integral of piece-wise function

On $R = [0, 1] × [0, 1]$, we define f(x, y) = { 1 if x is rational, 2y if x is irrational.} Show that $\overline{\int_R } f = 3/4$ and $\underline{\int_R}f = 5/4$. I'm confused about this question ...
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0answers
43 views

Riemann Upper and Lower sums of Heaviside function

I've hit a wall with heaviside functions, the question is to find the Riemann sums of, $$\int_{1}^{2}H(x-2)dx$$ and I've worked my way for the partition, $\Delta x = \frac{1}{n}$, hence $M_{i}=1+\frac{...
1
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1answer
84 views

Evaluating the limit : $\displaystyle \lim_{n \to \infty} \dfrac{1}{\sqrt{n}} \displaystyle \sum_{k=1}^n \dfrac{1}{\sqrt{n+k}}$

This kind of question always baffles me. It looks like the answer is 0 but it isn't. Can anyone tell me what does go on? And how do you evaluate this limit? $$\displaystyle \lim_{n \to \infty} \dfrac{...
1
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2answers
54 views

Understanding $\sum_{k=1}^n f(c_k)\Delta x_k$

Given function $f$ continuous on $[a,b]$, the area of the 'region' between the horizontal axis and the line described by $f$ can be approximated by the sum of multiple rectangles of equal width. This ...
0
votes
1answer
56 views

Help understanding Riemann sum to integral method in proof.

Given Abel's Identity according to Apostol [1] it follows that, $\sum_{y <n\leq y} a(n) f(n) = A(x) f(x) - A(y) f(y) - \int\limits_y^{x} A(t) f'(t) dt\quad\quad\quad$ where $A(x) = \sum_{n\leq y}...
0
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0answers
33 views

Volume of a solid as the limit of a sum and as an integral

I have this expression: $(x^2+y^2+z^2)^2=x^3+\frac{3}{10}x(y^2+z^2)$ I would like to calculate the volume as the limit of a sum and also as an integral. Can you also point out the difference ...
5
votes
2answers
73 views

Calculating integral using summation notation

I'm trying to understand $$\lim_{n\to\infty}\frac{3}{n}\sum_{k=1}^{n} \left(\left(\frac{3k}{n}\right)^2- \left(\frac{3k}{n}\right) \right).$$ I believe we can take $a=0, b=3$ and so this is equivalent ...
1
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0answers
30 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 ...
5
votes
1answer
121 views

Riemann sum of $x\cdot \ln(x)$

I did not find any information regarding this Riemann sum anywhere: Riemann sum of $f(x)=\begin{cases} 0& x=0 \\ x\cdot \ln(x)& \text{otherwise}\end{cases}$ in the interval $[0, 1]$. I don't ...
16
votes
0answers
336 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 $...
0
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1answer
63 views

Riemann sum $ \lim_{n\to \infty}\frac2n\sum_{k=1}^n \left(2+\frac kn\right)\left( \ln\left(2+\frac kn \right)\right)$ to integral

I don't know how to make the following limit $$\displaystyle \lim_{n\to \infty}\frac2n\sum_{k=1}^n \left(2+\frac kn\right)\left( \ln\left(2+\frac kn \right)\right)$$ into a definite integral and just ...