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Questions tagged [riemann-sum]

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

17
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1answer
1k views

Speed of convergence of Riemann sums

This question is inspired by a previous question. It was shown that, for all function $f \in \mathcal{C} ([0, 1])$, $$ \lim_{n \to + \infty} \sum_{k=0}^{n} f \left( \frac{k}{n+1} \right) - \sum_{k=0}...
5
votes
3answers
1k views

How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?

Prove that for any positive integer $n$, $$2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$$ Progress I think Riemann sum should be used for ...
10
votes
3answers
928 views

How do you calculate this limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{n^2+k^2}$?

How to find the value of $\lim_{n\to\infty}S(n)$, where $S(n)$ is given by $$S(n)=\displaystyle\sum_{k=1}^{n} \dfrac{k}{n^2+k^2}$$ Wolfram alpha is unable to calculate it. This is a question from a ...
13
votes
1answer
6k views

If a function is Riemann integrable, then it is Lebesgue integrable and 2 integrals are the same?

Is is true that if a function is Riemann integrable, then it is Lebesgue integrable with the same value? If it's true, how to prove it? If it's false, what is a counterexample?
4
votes
5answers
4k views

Proof that the area under a curve is the definite integral, without the fundamental theorem of calculus

Is there a proof that the area under a curve is equivalent to the definite integral, that doesn't involve the fundamental theorem of calculus. Perhaps a proof that uses Riemann sums.
14
votes
1answer
10k views

The absolute value of a Riemann integrable function is Riemann integrable.

This is an exercise in Bartle & Sherbert's Introduction to Real Analysis second edition. They ask to show that if $I=[a,b]$ is a closed bounded interval and that $f:I\to\mathbb{R}$ is (Riemann) ...
8
votes
2answers
369 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 ...
3
votes
1answer
654 views

If $f$ is Riemann integrable on $[0,1]$ then $\lim\limits_{n\rightarrow\infty}\frac{1}{n}\sum f\left(\frac{k}{n}\right)=\int\limits_{0}^{1} f(x)dx$ [closed]

I need to prove that if $f$ is Riemann integrable $[0,1]$ then $$\lim\limits_{n\rightarrow\infty} \frac{1}{n}\sum f\left(\frac{k}{n}\right)= \int\limits_{0}^{1} f(x)dx$$ My idea is to recognize the ...
1
vote
2answers
543 views

If $f$ is integrable then $|f|$ is also integrable

Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , \...
8
votes
2answers
322 views

Using right-hand Riemann sum to evaluate the limit of $ \frac{n}{n^2+1}+ \cdots+\frac{n}{n^2+n^2}$

I'm asked to prove that $$\lim_{n \to \infty}\left(\frac{n}{n^2+1}+\frac{n}{n^2+4}+\frac{n}{n^2+9}+\cdots+\frac{n}{n^2+n^2}\right)=\frac{\pi}{4}$$ This looks like it can be solved with Riemann sums, ...
9
votes
4answers
5k views

Is the indicator function of the rationals Riemann integrable?

$f(x) = \begin{cases} 1 & x\in\Bbb Q \\[2ex] 0 & x\notin\Bbb Q \end{cases}$ Is this function Riemann integrable on $[0,1]$? Since rational and irrational numbers are dense on $[0,1]$, no ...
10
votes
3answers
4k views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
5
votes
4answers
479 views

Solve $\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$ without using Riemann sums.

The natural way (and I think, the easier) to solve $$\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$$ involves Riemann sums. Multiplying and dividing by $n$, we get $...
6
votes
1answer
1k views

A Riemann integrable function must have infinitely many points of continuity

I was wondering whether anyone would be so kind as to briefly check my proof? I am supposed to prove the statement without using any theorems which would render the proof trivial. If $\displaystyle\...
7
votes
4answers
471 views

Convergence of a sequence (possibly Riemann sum)

Let $a_1, a_2, a_3, . . . , a_n$ be the sequence defined by $$ a_n = 2\sqrt{n}-\sum_{k=1}^{n}\frac{1}{\sqrt{k}} = 2\sqrt{n} - \frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-...-\frac{1}{\sqrt{n}} $$ show ...
5
votes
3answers
179 views

limit $\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$

How do I evaluate this? $$\lim_{n\to \infty }\sum_{k=1}^{n}\frac{k}{k^2+n^2}$$ I got concerned for that, I've tried make it integral for Riemann but it still undone.
0
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3answers
133 views

Riemann Sums proving function identically zero

Let $f$ be continuous on $[a,b]$. Suppose that $f(x)\geqslant 0$ for every $x \in [a,b]$ and that $$\int_a^b f(x)\mathsf dx = 0.$$ Prove that $f$ is identically zero on $[a,b]$. So I know to use ...
5
votes
3answers
3k views

Sum of divergent series

I saw a lot of article in Math SE like Why does 1+2+3+⋯=−1/12? and S=1+10+100+100+10000+…=−1/9? How is that and lot of others. Also I saw this one of Ramanujan summation but I do not get the ...
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 ...
4
votes
2answers
507 views

Is there any reason to expect the Riemann sum over $[a,b]$ to converge to the definite integral $\int_{a}^{b} f(x) \, dx$?

When learning the definite integral 'rigorously', most first courses seem to follow the steps below. Sketch the function over $[a,b]$ Construct arbitrary left and right function value partitions, ...
0
votes
2answers
2k views

Let $f(x)=x$ for $x\in[0,1]$ rational and $f(x)=0$ for $x\in[0,1]$ irrational. Prove that $f$ is not Riemann integrable on $[0,1]$.

Let $f:[0,1]\to\mathbb{R}$ where $f(x)=x$ for $x\in[0,1]$ rational and $f(x)=0$ for $x\in[0,1]$ irrational. Prove that $f$ is not Riemann integrable on $[0,1]$. I have a vague idea of what I'm ...
5
votes
0answers
2k views

The Lebesgue Criterion for Riemann Integrability — a proof without using the concept of oscillation.

I am trying to prove the Lebesgue Criterion for Riemann Integrability without using the concept of oscillation. The Lebesgue Criterion for Riemann Integrability states that if $ f: [a,b] \to \mathbb{...
4
votes
2answers
245 views

Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$.

Exercise: Suppose that $a<b$ and that $f:[a,b]\rightarrow R$ is continuous. Show that $\int_{a}^{c}f(x)dx = 0$ for all $c\in [a,b]$ if and only if $f(x) = 0$ for all $x\in [a,b]$. attempt of ...
4
votes
2answers
554 views

$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}$ using Riemann sums? [closed]

How to find the integral $$\int_{\frac{1}{3}\pi}^{\frac{2}{3}\pi} {\sin(x)\;dx}=1$$ using Riemann sums?
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 ...
0
votes
3answers
2k views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set of ...
2
votes
1answer
671 views

Prove Lipschitz function $f$ with constant $K$ is integrable on $[0, 1].$

We suppose that $f : [0, 1] \rightarrow \mathbb{R}$ is a Lipschitz function with constant $K$. We want to show that $f$ is integrable on $[0, 1].$ I've been trying to use the Darboux criterion of ...
1
vote
1answer
126 views

Sum of $\sum_{k=1}^n\sin{k\theta}$ [duplicate]

I have to calculate the sum of this series $$ \sum_{k=1}^n\sin{k\theta} $$ I tried solving it like this $$ \sum_{k=1}^n\sin{k\theta} = \operatorname{Im}\sum_{k=1}^ne^{(i\theta)k} $$ I recognized it as ...
1
vote
2answers
98 views

To evaluate the limits $\lim\limits_{n \to \infty} \{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}{n^3+n^3}\}$

To me it seems like that we need to manipulate the given sum into the Riemann sum of some function. First writing in the standard summation form; $$\{\frac{1}{1+n^3}+\frac{4}{8+n^3}+\ldots +\frac{n^2}...
1
vote
2answers
150 views

Given $f(x)$, a continuous function on [0, 1] st $f(x)≥0$ for all $x∈[0, 1]$, show that if $\int_0^1 f(x)dx=0$ then $f(x) = 0$ for all $x ∈ [0, 1]$ [duplicate]

Given $f(x)$, a continuous function on the interval [0, 1] such that $f(x) ≥ 0$ for all $x ∈ [0, 1]$, show that if $\int_0^1 f(x)dx = 0$ then $f(x) = 0$ for all $x ∈ [0, 1]$. Is this true if $f(x)$...
0
votes
1answer
88 views

Working with square roots

I'm asking for methods on working with and dealing with square roots in algebra. Numerous times square roots have been the bane of my existence and are usually difficult to get around. E.g. the ...
-4
votes
1answer
128 views

Riemann Lower Sum by definition [closed]

I have asked a similar question before, and didn't get the answer I was looking for, so i'll try to be more clear here. If necessary here is the link: Riemann sums upper and lower sums question Okay ...
21
votes
2answers
531 views

Does this sum of prime numbers converge?

$\newcommand{\P}{\operatorname{P}}$I'm wondering if this sum of prime numbers converges and how can I estimate the value of convergence. $$\sum_{k=1}^\infty \frac{\P[k+1]-2\P[k+2]+\P[k+3]}{\P[k]-\P[...
7
votes
7answers
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Finding limits using definite integrals $\lim_{n\to\infty}\sum^n_{k=1}\frac{k^{4}}{n^{5}}$

Find the limit of $\displaystyle\lim_{n\to\infty}\frac 1 {n^5}(1^4+2^4...+n^4)$ using definite integrals. It's equal to: $\displaystyle\lim_{n\to\infty} \sum^n_{i=1}\frac 1 i$ but now I'm not sure ...
5
votes
1answer
4k views

To prove the equivalence definition of Riemann integral.

I have some trouble with the Riemann integral, specifically, the definition of it in an article on wikipedia. We say that the Riemann integral of $f$ equals s if the following condition holds: For a ...
10
votes
2answers
628 views

Riemann sum on infinite interval

It is well known that in the case of a finite interval $[0,1]$ with a partition of equal size $1/n$, we have: $$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=0}^{n-1} f\left(\frac{k}{n}\right)=\int_0^...
7
votes
1answer
211 views

How to calculate the limit $\lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin(\frac{2k\pi }{n})$? Is it a Riemann sum?

I just came across this limit and I suppose it can be computed using a Riemann sum but I can't get it right. $$\lim _{n\to \infty }\sum _{k=1}^n\sqrt{n^4+k}\sin\left(\frac{2k\pi }{n}\right)$$ Any ...
4
votes
1answer
166 views

The limit of a sum

I'm trying to find out this limit $$\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{k=1}^n e^{-\Large\frac{k\pi\sqrt{2}}{4n}}\left(\tan\frac{k\pi}{4n}\right)^2 =?$$ My try: I know I have to transform the ...
6
votes
1answer
3k views

integration of 1/x as a riemann sum

To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: $$a, aq, aq^2, \ldots, aq^{n-1}, aq^n=b$$ where $q=\sqrt[n]{b/a}$. We then only need ...
5
votes
4answers
196 views

Evaluation of $\lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can write ...
3
votes
2answers
366 views

Use Darboux sums to calculate the area of $\sqrt x$ in $[0,1]$

I know that $$\Delta x=\frac{b-a}{n}=\frac{1-0}{n}=\frac{1}{n}$$ So $$\overline{D}=\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=1}^n\sup f(x_i)\Delta x=\displaystyle\lim_{n\to\infty}\...
1
vote
2answers
456 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.
0
votes
2answers
119 views

Verifying Riemann Sum

$$a_n = \frac{1}{n^2} \sum _{k=1}^n \left( \sqrt{\left(n+k\right)^2+n+k}\, \right)$$ $$\lim_{n\to \infty} a_n =\:?$$ I solved it like this and I would want to know it this makes sense and if not why....
8
votes
1answer
263 views

Question on Riemann sums

Question is : What I have read in Riemann sum definition is something like $$\sum_{i=1}^n f(y) .|x_i-x_{i-1}|$$ So, at first sight i am afraid this is not even related to Riemann integration of $f$ ...
6
votes
3answers
663 views

Calculate an integral with Riemann sum

We know that Riemann sum gives us the following formula for a function $f\in C^1$: $$\lim_{n\to \infty}\frac 1n\sum_{k=0}^n f\left(\frac kn\right)=\int_0^1f.$$ I am looking for an example where ...
5
votes
3answers
1k views

proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]

$$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$ when f is continuous on $[0,1]$ I know it can be proved using bounded convergence theorem but, I wanna know proof using only basic ...
4
votes
2answers
771 views

Understanding the definition of the Riemann Integral

In our real analysis course, we have been introduced to the idea of a Riemann Integral. I understand the intuitive concept of the Riemann Integral, but I don't quite understand the definitions of the ...
3
votes
2answers
1k views

Prove that If $f$ is integrable on $[a,b]$ and $[b,c]$ then $f$ is integrable on $[a,c]$

If $f$ is integrable on $[a,b]$ and $[b,c]$ then $f$ is integrable on $[a,c]$ and $$\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx .$$ Let's define three partitions $P,Q,R$ on $[a,c]$, $[a,b]...
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 $0$, $P_n=\{a=t_0^n<t_1^n<\ldots<t_{r_n}^n=b\}$, ...
3
votes
2answers
1k views

Show $\sin{\frac{1}{x}}$ for $x\not= 0$ and $f(0)=0$ is integrable on $[-1,1]$

Show $\sin{\frac{1}{x}}$ for $x\not= 0$ and $f(0)=0$ is integrable on $[-1,1]$. I guess my strategy for solving this is to use the following theorem: Let $f$ be a function defined on $[a,b]$. If $a&...