Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [riemann-sum]

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

3
votes
2answers
2k views

Showing that $f(x)=1$ if $x=\frac{1}{n}$, $0$ otherwise on [0,1] is Riemann Integrable

I have to show that the following function $f:[0,1]\rightarrow\mathbb{R}$ is Riemann Integrable: $$f(x) = \left\{ \begin{array}{ll} 1 & \mbox{if } x = \frac{1}{n} \\ 0 & \mbox{otherwise}...
0
votes
1answer
32 views

Limit as a definite integral (Riemann Sum)

I'm having a little trouble with a question that requires me to interpret a limit as a Riemann sum for an integral. However, I'm having trouble identifying which aspects of the limit correspond to the ...
0
votes
1answer
674 views

Integrals of sum. FInd upper and lower bounds

Find the upper and lower bound using integrals. $$\sum_{k=1}^n (k^2 - 3k)$$ Please explain I actually want to understand it
0
votes
0answers
70 views

Laplace Transform: Continuous analogue of Power series

Laplace transform is considered as the continuous analogue of the power series, $$A(x)=\sum_{n=0}^\infty a(n)x^n \rightarrow A(x)= \int_0^\infty a(t)x^t\mathbf {dt} $$ sub $\,\, x^t=e^{(\ln\,x)^t} $ ...
0
votes
4answers
56 views

Why do we use Riemann approximations when we can find actual area by using integrals

I am a calculus 1 student. I was wondering that if Riemann sums only give us an approximation(either over-estimate or under-estimate) the area under the curve, Why do we celebrate Riemann sums(...
0
votes
0answers
15 views

Calculating upper Riemann sum. Is it sufficient to only consider “simple” partitions

Let $b > 0$. I'd like to calculate $ \int_{0}^b x^2 dx$ using upper and lower Riemann sums. Since the function is continuous on $[0,b]$ I know that it is integrable so I only need to calculate $$ ...
3
votes
1answer
61 views

Question about Riemann integrability: do we need to specify that all Riemann sums converge to the same number in the definition?

Let $f:[a,b]\rightarrow\mathbb{R}$ be a function. Suppose that there is a sequence of partitions $\{P_n\}_{n=1}^\infty$ with mesh tending to $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
1
vote
1answer
83 views

Find $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left(e^{(1+\frac{k}{n})^2} - \frac{3e^{(1 + \frac{3k}{n})}}{2\sqrt{1 + \frac{3k}{n}}}\right).$

Find $$\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \left(e^{(1+\frac{k}{n})^2} - \frac{3e^{(1 + \frac{3k}{n})}}{2\sqrt{1 + \frac{3k}{n}}}\right)$$ Choices: A: 1 B: $\frac{1}{2}$ C: $\frac{1}{...
0
votes
1answer
173 views

Using summation notation to solve Riemann Sum problems

I am learning how to write summation notation to solve Riemann Sum problems. I have a function $f(x)=2x^3 -5x^2 +9x -5$ and I need to find the area under the curve with 8 intervals for $-4 \leq x \...
0
votes
1answer
26 views

trying to proove this limit equality to a specific integral [closed]

can you help me prove this equality? I tried to use Riemann sums but I haven't succeeded to find something useful. $$\lim_{n\to\infty} \sum_{k=1}^n f\left(\frac{k}{n}\right)\frac{1}{n} = \int_0^1f(...
1
vote
1answer
99 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
3answers
60 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}...
6
votes
5answers
360 views

Compute the following limit, possibly using a Riemann Sum

$$\lim _{n\to \infty }\sum _{k=1}^n\frac{1}{n+k+\frac{k}{n^2}}$$ I unsuccessfully tried to find two different Riemann Sums converging to the same value close to the given sum so I could use the ...
0
votes
1answer
50 views

What is the rigor behind U subsitution?

$ \int f(g(x)) dx = \int \frac {f(u)}{u'} du$ requires that $ \int f(x) dx = \int f(x) \cdot dx$ but dx just represents the variable that F(x) +c is a function of. So why is it legal for dx to be ...
0
votes
0answers
37 views

How much information about a function do you need to determine the function?

I am messing around with a program that computes definite integrals using Riemann sums and had questions about constructing graphs and determining functions. So let's say my program gives me any 20 ...
0
votes
2answers
33 views

Finding a limit using Riemann sum

In the interval [0,1] I have to find the limit of a Riemann sum $$\lim _{n\to \infty }\sum _{i=1}^n\left(\frac{i^4}{n^5}+\frac{i}{n^2}\right)$$ so far I have this $$\lim _{n\to \infty }\sum _{i=1}^n\...
3
votes
1answer
2k views

Riemann Sum Approximations: When are trapezoids more accurate than the middle sum?

We can approximate a definite integral, $\int_a^b f(x)dx$, using a variety of Riemann sums. If $T_n$ and $M_n$ are the nth sums using the trapezoid and midpoint (middle) sum methods and if the second ...
1
vote
1answer
54 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;...
2
votes
1answer
67 views

An Issue about Riemann Integral

By the denition of Riemann integral, we have $$\lim_{n\rightarrow \infty}\sum_{k=1}^{n}f(a + \frac{b-a}{n}k)\frac{b-a}{n} = \int_{a}^{b}f(x)dx$$ In a particular problem, for $k<n$, $f$ is defined, ...
1
vote
0answers
64 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
46 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
86 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&...
2
votes
1answer
936 views

Trouble integrating 1/x from Riemann Sum

Preface: I'm a A-Level student, so much of the maths I'm speaking about here is quite new to me, in particular Riemann Sums. I apologise if this already has an answer, I couldn't find it. I'm trying ...
3
votes
1answer
55 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
votes
1answer
1k views

Evaluate the Riemann sum

If $\mathrm{f}\left(x\right) = 2\cos\left(x\right)$ $ 0 \leq x \leq 3\pi/4$ evaluate the Riemann sum with $n = 6$, taking the sample points to be left endpoints. ( Round your answer to six ...
0
votes
0answers
36 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 ...
7
votes
4answers
479 views

How to integrate $xe^x$ without using antiderivatives or integration by parts.

Yesterday, I sat for my Real Analysis II paper. There I found a question asking to integrate $\displaystyle\int_0^1 xe^x \, dx$ without using antiderivatives and integrating by parts. I tried it by ...
1
vote
0answers
39 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 ...
1
vote
1answer
90 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{...
3
votes
2answers
511 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}}}...
4
votes
0answers
80 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 ...
1
vote
2answers
73 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
52 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
53 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 ...
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
213 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
votes
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
vote
1answer
98 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
votes
1answer
47 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
vote
3answers
77 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
1answer
1k views

Finding the lower and upper Riemann Sums

I'm having a hard time figuring this questions out. I've looked on google and on the book and so far I haven't gotten a good explanation for this questions. I know they aren't hard and are probably ...
1
vote
1answer
74 views

How can you use combinations to find a general formula for the power series summation

I do not understand the fundamental theorem of calculus so I have been trying to find a proof that the antiderivative of a function evaluated at $x=b \text{ and } x=a$ gives the area under the curve ...
0
votes
0answers
20 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
43 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
63 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
36 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\...
12
votes
1answer
326 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 ...
0
votes
1answer
46 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}...