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Questions tagged [riemann-sum]

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

3
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1answer
57 views

Finding $\lim\limits_{x\to1^-}\Bigl(\prod\limits_{n=0}^{\infty}\Bigl(\frac{1+x^{n+1}}{1+x^n}\Bigr)^{x^n}\Bigr)$

Compute : $$L=\lim_{x\to 1^-} \left(\prod_{n=0}^{\infty} \left(\frac {1+x^{n+1}}{1+x^n}\right)^{x^n}\right)$$ My try: Writing out the first few terms I noticed that the limit can be expressed as $$...
2
votes
0answers
16 views

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions?

Is it necessarily the case that $\lim_{k\to\infty} \text{gap } P_{k} = 0$ if $\{P_{k}\}$ is an Archimidean sequence of partitions? I know that by the definition of an Archmedean sequence of ...
0
votes
0answers
20 views

Riemann Integrable definition in partition distingued $\mathbb{Q}$

A function $f:[a,b]\rightarrow \mathbb{R}$ is Riemann Integrable on $[a,b]$ if $\exists \thinspace L \in \mathbb{R} : \forall \epsilon >0\thinspace \exists \thinspace \delta >0 :$ if $P$ is any ...
2
votes
1answer
48 views

Function is Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$

i need to prove that if Riemann integrable in $[a,c]$ and $[c,b]$ then is RI in $[a,b]$, maybe is easy but i can't see it. Definition of Riemann Integral A function $f:[a,b]\rightarrow \mathbb{R}...
0
votes
1answer
35 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
0answers
71 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
57 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
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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
62 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
84 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
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(...
0
votes
1answer
51 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
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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
34 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\...
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}...
1
vote
1answer
55 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
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&...
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
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 ...
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 ...
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}}}...
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 ...
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 ...
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
262 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
99 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
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
44 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
66 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
43 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
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}...
1
vote
1answer
103 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
46 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
vote
2answers
82 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
72 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
328 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
97 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 $|...