All Questions
Tagged with riemann-sum limits
242 questions
5
votes
2
answers
199
views
Prove That $\int_{1}^{e}\frac{1}{t}dt=1$ Without the Use of Logarithms
I've been studying Real Analysis by Jay Cummings, and am working through the exercises on integration. The question is as such:
Define a function $L:(0, \infty)\rightarrow\mathbb{R}$ by $L(x)=\int_{1}^...
- 148
2
votes
1
answer
84
views
Asymptotic analysis of the finite product
I have the following product
$$\prod _{n=1}^{\frac{L}{2}} \frac{e^{-\frac{2 \cos \left(\frac{\pi n}{L+1}\right)}{T}}+1}{e^{-\frac{2 \cos \left(\frac{\pi n}{L}\right)}{T}}+1}$$
I am interested in ...
- 734
0
votes
2
answers
79
views
Calculate this limit using integral
Giving $a$, $b$ are two positive numbers and $\alpha$ is a positive integer; calculate the limit:
$$
\lim_{n\to\infty}\left[\dfrac{1}{n^{2\alpha+1}}\sum_{k=1}^{n}\left(k^2+ak+b\right)^{\alpha}\right]
$...
- 159
1
vote
2
answers
59
views
How to calculate the limit of this riemann sum using integrals?
$$
\lim_{n\to \infty} \frac{3}{n} \sum_{k=1}^{n} \sqrt{\frac{n}{n+3(k-1)}}
$$
Assuming this can be written as a Riemann Sum, how can I bring it to an integral?
I'm trying to make it reach the form $\...
- 375
3
votes
3
answers
80
views
Doubt regarding limits on riemann sums
Find $$\lim_{n\to\infty}\frac{1^p+3^p+\ldots+(2n+1)^p}{n^{p+1}}$$
I found a solution here which goes like this:
By Riemann sums, for any $p>-1$:
$$ \frac{1}{n}\sum_{k=0}^{n}\left(\frac{2k+1}{n}\...
- 565
0
votes
0
answers
99
views
Is $\lim\limits_{n \to\infty}\sum_{i=1}^{l\cdot n}\frac{f\left(\frac{i}{n}\right)}{n}$ a valid definite integral riemann sum? What is it called if so?
I came up with this alternate Riemann sum that correctly gives the value of a definite integral (at least for some simple polynomial and trig functions I tested with wolfram alpha):
$$\lim\limits_{n \...
1
vote
1
answer
54
views
Limit of $∞.0$ form of an integral and Riemann sum
I was pondering how to solve a limit of the kind $$\lim_{n \to ∞} n^k \left(\dfrac{1}{n}\sum_{k=0}^{n-1} f \left(\dfrac{k}{n} \right) -\int_0^1 f(x)dx \right)$$
where k is chosen such that the order ...
- 725
9
votes
1
answer
740
views
Calculating pretty difficult limit that invloves Riemann sums
Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit
$$\lim_{n \to \infty} n\left(n\Big(\ln(1+\sqrt{2})-S_n\Big)-\frac{1}{2\sqrt{2}\,(1+\sqrt{2})}\right).$$
My intuition ...
- 1,196
3
votes
2
answers
113
views
Convergence of a sum as limit tends to infinity that seems to be harmonic series
I have come across a mathematical problem that is to evaluate the expression:
$$
lim_{n\rightarrow\infty} \left\{\frac{1}{\sqrt{2n-1^2}}+\frac{1}{\sqrt{4n-2^2}}+\frac{1}{\sqrt{6n-3^2}}+...+\frac{1}{\...
- 1,483
0
votes
0
answers
265
views
Understanding the Definition of Indefinite Integral Using Riemann Sums
The definite integral of a function $f$ from $x=a$ to $x=b$ and $\Delta x = (b-a)/n$ is defined by the limit of a Riemann Sum:
$$
\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(a+i\cdot\...
-1
votes
2
answers
89
views
Find a definite integral which represent $ \lim_{n\to\infty}{\frac{1}{n}\sum_{k=1}^{n}{\sqrt\frac{k}{n+k}}} $
Find a definite integral which represent $$\lim\limits_{n\to\infty}{\frac{1}{n}\sum\limits_{k=1}^{n}{\sqrt\frac{k}{n+k}}}.$$
I don't know how can I approach the question.
Is the answer $$\int_{0}^{1}{\...
1
vote
0
answers
59
views
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 $L=\lim_{n\to\infty}\left(\prod_{k=0}^n \frac{2n+2k}{2n+2k+1}\right)$
$$\ln(L)=\lim_{n\to\infty}\left(...
- 6,781
0
votes
0
answers
81
views
Prove the following definite integrals as limits of sum $\int_a^b \frac{1}{x^2}dx = 1/a - 1/b$ [duplicate]
My attempt so far ..
$\int_a^b f(x)dx = \lim \sum_{1}^{n} h f(a+rh) $, where n tends to infinity, $nh = b-a$
Here $f(x) = \frac{1}{x^2}$ , using this we get
$\int_a^b f(x)dx = \lim \sum_{r=1}^{n^2} h ...
4
votes
2
answers
212
views
Evaluate the limit $\lim_{n\to \infty }\sum_{i=1}^n \frac{1}{n} \cdot \lfloor \sqrt {\frac{4i}{n}} \rfloor$
I solved the problem using the Riemann integral. However, my answer did not match with the result given in the book. My answer was $\frac{3}{4}$ and the answer given in the book was just 3.
Help me ...
- 91
0
votes
1
answer
119
views
Limit of Infinite Sum using Riemann sum
I was trying to learn about finding the limit of an infinite series using Riemann sums and I derived the following conclusion using the basic Riemann definition of definite integration:
$$\int_{0}^{k}...
- 333
3
votes
3
answers
176
views
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 ...
- 41
1
vote
0
answers
21
views
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 ...
- 11
2
votes
4
answers
141
views
Prove $\lim \limits_{h \to 0} \frac{a^h-1}{h} = \ln (a)$ without using L'hospital's.
I'm a Calc 2 student and was curious as to why $\frac{d}{dx}a^x = a^x\ln a$. Using the limit definition you can arrive at $\frac{d}{dx}a^x = \lim \limits_{h \to 0} \frac{a^x(a^h-1)}{h}$ so the part ...
0
votes
1
answer
92
views
Clever ways to expand $\prod_{i=1}^{n}\left(1+\frac{n^2}{i^2}\right)\left(1+\frac{i^2}{n^2}\right)^\frac{n^2}{i^2}$?
This is a continuation from this thread.
From my work in the link above I found the following.
$$\int_{0}^{\infty}\ln\left(1+\frac{1}{x^2}\right)dx=\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^n\ln\left(\...
- 2,479
7
votes
3
answers
377
views
Improper Integrals as Riemann Sums and a Beautiful Limit $\lim_{n \to \infty}\frac{\sqrt[n]{n!}}{n}$
For some context, I recently encountered a beautiful limit.
$$\lim_{n \to \infty}\frac{\sqrt[n]{n!}}{n}$$
To solve this we begin by taking the natural log of the inside.
$$\ln\left(\frac{\sqrt[n]{n!}}{...
- 2,479
6
votes
0
answers
96
views
Limit of $2^{n^2/2}\sum_{j=1}^{n/2} \sum_{k=1}^{n/2}\left(\cos^2(\frac{j \pi}{n+1}) + \cos^2(\frac{k \pi}{n+1})\right)$ as a double integral
I am currently looking into Dimer coverings and my next step is to find how the following limit is calculated:
$$\begin{align*}
L &= \lim_{n \to \infty}\frac{1}{n^2}\ln\left(2^{n^2/2}\prod_{j=1}^{...
- 139
2
votes
1
answer
286
views
Riemann sum of infinite series
Let $f$ be a non-negative, bounded and continuous function such that $\int_\mathbb{R} f(x)\, \mathrm{d}x < \infty$.
Does it hold that
$$
\lim_{n \rightarrow \infty} \sum_{j \in \mathbb{Z}} \frac{1}{...
- 279
3
votes
1
answer
61
views
How to solve $\lim\limits_{n\to∞}\frac1{n^{3/2}}(\sqrt{2n+1}+\sqrt{2n+2}+\cdots+\sqrt{2n+n})$
How to solve $\lim\limits_{n\to∞}\dfrac1{n^{3/2}}(\sqrt{2n+1}+\sqrt{2n+2}+\cdots+\sqrt{2n+n})$
This problem was asked by another user, but was deleted when I found the answer below. So I thought I ...
- 539
3
votes
2
answers
109
views
Evaluate the limit of a sequence by Riemann sums and mean value theorem
Calculate:
$\displaystyle \lim_{n \rightarrow \infty} \left( \frac{n \pi}{4} - \left( \frac{n^2}{n^2+1^2} + \frac{n^2}{n^2+2^2} + \cdots \frac{n^2}{n^2+n^2} \right) \right)$.
I solved it by taking ...
- 115
1
vote
1
answer
84
views
How do we find $\lim_{n \to \infty} \sqrt{n}(1 - \sum_{k=1}^n 1/(n + \sqrt{k}))$? [closed]
$$\lim_{n \to \infty} \sqrt{n}\left(1 - \sum_{k=1}^n \frac{1}{n + \sqrt{k}}\right)$$
Guys, please help to solve the limit.
I have try transfer it to integrate problem, but it works not good. My ...
- 27
1
vote
0
answers
60
views
How do I prove that this finite sum evaluate to $1+\sqrt{2}$ for all values of n? [duplicate]
$$
\frac{\displaystyle\sum_{i=1}^{n-1}
\sqrt{\sqrt{n} + \sqrt{i}}}
{\displaystyle\sum_{i=1}^{n-1}
\sqrt{\sqrt{n} - \sqrt{i}}}
= 1 + \sqrt{2}
$$
When $n=2$, this is easy to verify. As $n \to \infty$...
- 1,240
1
vote
1
answer
78
views
Is the statement $\sum_{j=1}^\infty x_j<\infty,~(x_j\ge0)$ $\Longrightarrow \lim _{k \to \infty} \sum_{j=k}^\infty x_j=0$ true?
As the title states, I would like to know if the statement
$$
\sum_{j=1}^{\infty} x_{j}<\infty \Longrightarrow \lim _{k \to \infty} \sum_{j=k}^{\infty} x_{j}=0,\qquad x_j\in [0,\infty)
$$
is always ...
- 853
4
votes
6
answers
289
views
Is it possible for Riemann Sum and Standard Integration to have different answers?
I have a specific question for this equation :
$$\frac{1}{x+1}
$$
Using Standard Integration,
$$\int_{0}^{1} \frac{1}{x+1}
$$
which is approximately 0.69
Using Riemann Sum (right end point), however, ...
- 157
3
votes
2
answers
138
views
Evaluating limit using Riemann sums
I am preparing for calc II exam, and i have some trouble with 2 problems.
$$ \lim_{n \to \infty} \frac{1}{7n^2}+\frac{1}{7n^2+1}+\frac{1}{7n^2+2}+ \dots + \frac{1}{8n^2}$$
$$ \lim_{n \to \infty} \sum_{...
- 474
1
vote
1
answer
45
views
Finding the limits while changing limit of an infinite sum into integral.
I was solving the following question.
Find the following limit.
$$\lim_{n\to \infty}\dfrac1n \left(\dfrac{1}{1 + \sin\left(\dfrac{\pi}{2n}\right)} + \dfrac{1}{1 + \sin\left(\dfrac{2\pi}{2n}\right)} + ...
user983206
-2
votes
1
answer
46
views
Representing the area of a circle, with radius $1$ as the sum of inscribed circumferences $\lim_{n\to\infty} \sum_{k=1}^n \frac{k}{n}\cdot 2\pi$ [closed]
I saw that a similar question had been asked, but mine is specifically about what is wrong with this sum representation.
So, if we were to imagine a circle of radius 1, that radius can be divided into ...
4
votes
1
answer
154
views
How to take limits of 'almost Riemann' sums like $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} \cos (a \pi k \log(n)/n)$
How can I solve limits of sums that are 'almost' Riemann, but can't be written in the typical form (i.e, $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} f(k/n)$ which we can rewrite as an integral $\...
0
votes
0
answers
32
views
Limit of partial Harmonic Sum as a bounded integral
Can anyone explain why $(b,a)=(1,0)$ where $b-a=1$.
$\lim_{n\to \infty}(\frac{1}{n+1} +\frac{1}{n+2} +...+\frac{1}{n+n})=\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n \frac{1}{1+i/n}=\int_a^b \frac{dx}{1+...
- 102
-3
votes
2
answers
140
views
Conventionally $\int_2^0 f(x)dx:=-\int_0^2 f(x)dx$ whereas $\sum_{i=2}^0 := 0$ Why are definite integrals and series treated differently?
Integrals are defined in terms of series, so why is the treatment of definite integrals different than the treatment of series when the lower limit is greater than the upper limit.
For a definite ...
- 1,517
1
vote
1
answer
73
views
$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k+\theta}{n})=\int_{0}^{1}f(x)dx$ where $\theta\in(0,1)$?
Let $f\colon [0,1]\to\mathbb{R}$ be a continuous (or Riemann-integrable) function. As we already know, the next equation holds:
$$
\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f\left(\frac{k}{n}\right)=...
- 113
0
votes
1
answer
50
views
$\lim_{n \to \infty} {\frac{1}{n^2}\sum_{k=0}^{n}{\frac{1}{\ln{(1 + \frac{(n+k)\sqrt{n^2+k^2}}{n^3})}}}}$
How can I solve
$$\lim_{n \to \infty} {\frac{1}{n^2}\sum_{k=0}^{n}{\frac{1}{\ln{(1 + \frac{(n+k)\sqrt{n^2+k^2}}{n^3})}}}}$$
It looks like a Riemann limit to me, but I'm not able to get it to a final ...
- 1,886
0
votes
2
answers
58
views
$\int_a^b e^x dx$ using limits.
$$\int_{a}^{b}{e^{-x}}dx$$
by definition
$$b-a=nh$$
$$\int_{a}^{b}{e^{-x}} dx = \lim_{h \to 0}{he^{-(a)}+he^{-(a+h)}+he^{-(a+2h)}+...+he^{-(a+(n-1)h)}}$$
which can further be simplified to
$$\int_{a}^{...
- 345
0
votes
1
answer
89
views
Limit of a Riemann Sum. [duplicate]
I am trying to calculate the limit
$$\lim_{n\to\infty}\sum_{k=n+1}^{2n}\frac{1}{k}$$
Can someone please explain how I can go about doing this?
- 31
3
votes
1
answer
282
views
Find limit $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$.
Find the following limit:
$$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^4}{k^5+n^5}$$
I had an idea of using upper Riemann sum for function $x^4$ on interval $[0,1]$ but I don't know how to deal with $...
- 41
0
votes
1
answer
148
views
Evaluate a limit in combinatorics [duplicate]
Evaluate $$\lim_{n\rightarrow \infty} \sum_{r=0}^n \frac{1}{n\choose r}.$$
I tried it using the Reimann sum(turning it into a definite integral) but it got me confused with the factorials in it. I ...
0
votes
1
answer
76
views
$x \lim_{n\to\infty}\sum^n_{r=1}\frac{1}{n}\ln\left(\frac{1+\frac{xr}{n}}{1+\frac{x²r²}{n²}}\right)$ into an integral.
$$x \lim_{n\to\infty}\sum^n_{r=1}\frac{1}{n}\ln\left(\frac{1+\frac{xr}{n}}{1+\frac{x²r²}{n²}}\right)$$ into an integral.
So this beast right here is the result of a bigger one: $$\lim_{n\to\infty}\...
- 487
0
votes
1
answer
95
views
What's going on with this infinite product?
Define, with $x$ real valued:
$$
F_N(x) = \prod_{k=0}^N\left|x-\frac{k}{N}\right|^{1/N}
$$
We can calculate the limit for $N\to\infty$ as a Riemann sum converging to an integral:
$$
F_\infty(x) = \...
- 17.3k
1
vote
1
answer
95
views
Having trouble with finding a limit for this sequence $a_n=\frac{n+1}{n}\ln(\sqrt{n})-\frac{1}{n^2}\sum^n_{k=1}\ln(k+n)^k$
We got this question as a bonus for our homework assignment and
I'm having trouble figuring out how to start to solve this limit. We need to find $\lim_{n\rightarrow\infty}(a_n)$ for this sequence:
$$...
- 15
0
votes
0
answers
37
views
Possibles ways to define the Riemann integral
Can you recommend some nice papers or texts that discuss in detail the different ways to define Riemann integrability (for bounded functions defined on a compact interval of the real line)?
For ...
- 507
4
votes
2
answers
275
views
Using definite integral to find the limit.
I would like someone to verify this exercise for me. Please.
Find the following limit:
$\lim\limits_{n \to \infty}\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n}\right)$
$=\lim\limits_{n \to \...
- 1,161
2
votes
2
answers
72
views
Evaluating $\lim_{x\to 0} \left(\sum_{r=1}^{n}r^{1/\sin^2x}\right)^{\sin^2x}$
$$\lim_{x\to 0} \left(\sum_{r=1}^{n}r^{\frac{1}{\sin^2x}}\right)^{\sin^2x}$$
$$\begin{aligned}\lim_{x\to 0}\left(\sum_{r=1}^{n}r^{\frac{1}{\sin^2x}}\right)^{\sin^2x}&=n\lim_{x\to 0}\left(\sum_{r=...
- 6,431
0
votes
2
answers
192
views
Can we use limit of Riemann Sum? [closed]
Let $$\alpha = \lim_{n\to\infty}\sum_{m = n^2}^{2n^2}\dfrac{1}{\sqrt{5n^4 + n^3 + m}}.$$ Then what is $10\sqrt5\,\alpha$ is equal to?
I am trying to solve this problem as the limit of Riemann sum and ...
- 75
7
votes
2
answers
154
views
Help with $ \lim \frac{n\pi}{4} - \left( \frac{n^2}{n^2+1^2} +\frac{n^2}{n^2+2^2} + \cdots + \frac{n^2}{n^2+n^2} \right) $
I have proved that $$\lim_{n\to \infty} \left( \frac{n}{n^2+1^2} +\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} \right) = \frac{\pi}{4},$$
by using the Riemann's sum of $\arctan$ on $[0,1]$. Now I'...
- 1,271
1
vote
2
answers
151
views
A quite Riemann sum with square root
I would like to compute the following limit:
$$\lim_{N\to\infty}\frac{1}{N^2}\sum_{i,j=0}^{N-1} \sqrt{i+1}\sqrt{j+1}.$$
It's looks like a Riemann sum minus a factor $\frac{1}{N}$ is the square root. ...
- 95
4
votes
1
answer
157
views
Calculate the Limit of Double Sum
Compute
\begin{equation}
L=\lim _{n \rightarrow \infty}\frac{1}{n} \sum_{a=1}^n \sum_{b=1}^n \frac{a}{a^2+b^2 }.
\end{equation}
My attempt:
Define \begin{equation} f(n,m)= \frac{1}{n} \sum_{a=1}^n \...
- 665