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Questions tagged [riemann-sum]

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

2
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2answers
49 views

Limit of the sum using integral

$\lim\limits_{n\rightarrow\infty}\sum_{k = 1}^{n} \frac{1}{(k+n)\sqrt{1 + n\ln({1+\frac{k}{n^2}})}}$. I can find it using integral: $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum_{k = 1}^{n} \frac{1}...
1
vote
1answer
46 views

A property of Darboux sum

I'm trying to show that: $$\overline{I}:=\inf _P S(f;P)=\lim_{\lambda (P)\to 0}S(f;P)$$ where $P$ is a generic partition (made by $n$ points) of the interval $[a,b]$, $f$ is bounded on $[a,b]$, $S(f;...
1
vote
0answers
56 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 ...
1
vote
3answers
42 views

How to find limit of sum $\lim\limits_{n\to\infty}\sum_{k=1}^{100n}\frac{k^p}{n^{p+1}}$

How can I find this limit? I've tried to use Stolz theorem, but have not succeed. I have heard smth about Riemann sums, but have not found good algorithm how to use it. Can you help me to solve it ...
2
votes
1answer
80 views

Find the limit of $\lim\limits_{n\to\infty}\frac{1}{n^2}\sum\limits_{k=1}^{n}k\arctan{\big(\frac{pk-p+1}{pn}\big)}$

I am required to find the limits of two "siblings" using the same idea, they are: $$\lim_{n\to\infty}\sin{\left(\frac{1}{pn+r}\right)}\sum_{k=1}^{n}\sin{\left(\frac{2pk-2p+r}{2pn}\right)}$$ with $0&...
3
votes
1answer
53 views

proof that there is $c \in [a,b]$ such that $f(c) = g(c)$

Let $f,g: [a,b] \rightarrow \mathbb{R}$ continuous functions such that $\int_a^{b} f(x)dx = \int_a^{b}g(x)dx$. Proof that there is $c \in [a,b]$ such that $f(c)=g(c).$ This questions has been asked ...
0
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0answers
35 views

Approximate the area under the curve $f(x) = x^2+4x+6$ on the interval $[2,6]$ using the right-hand Riemann sum

Approximate the area under the curve $f(x) = x^2+4x+6$ on the interval $[2,6]$ using the right-hand Riemann sum where $P$ is the partition of $[2,6]$ determined by $\{2,4,5,6\}$ I set up the right ...
1
vote
0answers
38 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 ...
3
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2answers
509 views

Is this a Riemann sum (if so, I can't figure out which one)?

This was supposedly an easy limit, and it is suspiciously similar to a Riemann sum, but I can't quite figure out for what function. $$\lim_{n\to\infty}{\frac{1}{n} {\sum_{k=3}^{n}{\frac{3}{k^2-k-2}}}...
1
vote
2answers
70 views

Evaluate $\lim_{n\to\infty} \sum_{i=1}^{n} \left[\frac{4}{n} - \frac{i^2}{n^3}\right]$

$$\lim_{n\to\infty} \sum_{i=1}^{n} \left[\frac{4}{n} - \frac{i^2}{n^3}\right]$$ If I take out $\frac{1}{n}$, it looks like a Riemann sum: $$\lim_{n\to\infty} \sum_{i=1}^{n} \left[\left(\frac{1}{n}\...
2
votes
2answers
51 views

Evaluate Integral $\int_{1}^{3}(3x + 2)dx$ as a Riemann Sum

I am having difficulties working out this Riemann Sum. $$ \Delta x = \frac{2}{n},~~ x_i = 1 + \frac{2i}{n},~~ i = \frac{n(n+1)}{2}$$ where $$\int_{1}^{3}(3x + 2)dx = \lim_{n\to\infty} \sum_{i=1}^{n} ...
2
votes
1answer
48 views

Understanding what ij mean in a Double Riemann Sum (Double Integral)

I am having trouble understanding what the ($x^*_{ij} $, $y^*_{ij}$) in this diagram (circled in blue) is explaining. What I do know is that $i$ is the iteration of the $x$ Riemann Sum and the $j$ is ...
4
votes
0answers
71 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 ...
0
votes
1answer
52 views

Evaluating an Integral as a Riemann sum

Evaluate the integral as a Riemann sum $\int_{0}^{2} 4x^3dx$. My book defines an definite integral as $$ \int_{a}^{b} f(x) dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i) \Delta x $$ where ${x_i} = a+ ...
0
votes
1answer
63 views

Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$.

Evaluate the limit by first recognizing the sum as a Riemann Sum for a function defined on $[0,1]$ $$ \lim\limits_{n \to \infty} \sum\limits_{i=1}^{n} \left(\frac{i^5}{n^6}+\frac{i}{n^2}\right). $$ I ...
0
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2answers
53 views

Series interpretation of integral

I'm currently stuck with the following question: Prove, that $\ln(2) = \lim\limits_{n \to \infty} \sum_{k=1}^n \frac{1}{n+k}$ by rewriting the left side as an Integral. So my current thoughts are: $...
1
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1answer
94 views

Cosine of a Wiener process

Let $W_t$ be a standard Brownian motion, i.e., $W_t \sim N(0,t)$. Define the random variable $$X=\int_0^1\cos(W_t)dt$$ A similiar process, $Y_t=\cos(\omega t+\sigma W_t+\theta)$, with the uniform ...
-1
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1answer
44 views

What are the requirements of a function so that the left Riemann sum equals the right Riemann sum?

My homework question in particular specifies over an interval of [0,1], the function is negative, and the function is decreasing.
1
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3answers
72 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}{...
0
votes
0answers
19 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
votes
2answers
41 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
votes
2answers
61 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} &...
1
vote
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
vote
1answer
30 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\...
0
votes
1answer
45 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
votes
1answer
66 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}...
0
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0answers
73 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
votes
1answer
40 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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
33 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 ...
1
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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\{\...
0
votes
1answer
42 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
votes
2answers
70 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
votes
1answer
41 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
votes
1answer
325 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
vote
1answer
68 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
votes
1answer
96 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
votes
1answer
44 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
votes
1answer
44 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
vote
0answers
28 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^...
0
votes
0answers
42 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$_{i}$]) - ...
0
votes
0answers
54 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 ...
0
votes
0answers
63 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
votes
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
votes
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 ...
0
votes
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, ...
-1
votes
2answers
32 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?
1
vote
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 ...
1
vote
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 ...
0
votes
2answers
57 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 ...