# Questions tagged [riemann-sum]

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

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### Working on details on the Secretary Problem

I've been trying to follow this proof of the optimal way to solve the secretary problem (ref. https://en.wikipedia.org/wiki/Secretary_problem). Everything is clear to me except where they are ...
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### A question about sum of sequence with fifth powers

$\sum_{r=1}^{p}(4p+3r)^5$ I'm looking for the coefficient of the highest degree term in the formula obtained when this sum is written in terms of $p$. Is there a practical way to do this? And also ...
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### Solving Sequence Using Riemann Sum

I am currently in my second calculus course and my professor asked me to evaluate the limit of a sequence. $$b_k = \frac{1}{9k+1} + \frac{1}{9k+2} + \cdots + \frac{1}{20k}$$ We did a similar problem ...
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### Is this a Riemann sum?

I have come a cross with a sum that looks like this: $$\sum_{x\in{\Lambda_N}}\epsilon^2 k(\epsilon x)e^{-i\pi\omega \cdot \epsilon^2 x}\quad \quad\quad\quad(*)$$ Here $x$ takes values in the discrete ...
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### Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
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### Let $f:[0,1]\to \Bbb R$ such that $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$

A function $f$ is defined on $[0,1]$ by $f(x)=x$ if $x$ be rational $x^2$ if $x$ be irrational. Find $\underline{\int}_0^1 f$ and $\overline{\int}_0^1f.$ The solution given is as follows: $f$ is ...
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### 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 ...
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### How to perform this sum

I encountered this sum $$S(N,j)= \frac{2 \sqrt{2}h(-1)^j}{N+1}\cdot\sum _{n=1}^{\frac{N}{2}} \frac{\sin ^2\left(\frac{\pi j n}{N+1}\right)}{\sqrt{2 h^2+\cos \left(\frac{2 \pi n}{N+1}\right)+1}},$$ ...
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### How to prove if $f$ is Darboux integrable then for all $\epsilon > 0$ then $U(f, P_{\epsilon}) - L(f, P_{\epsilon}) < {\epsilon}$ ??

Background: I am studying Real Analysis (never studied it before) from the book 'Real Analysis' by Jay Cummings. I am at chapter 8 (Integration) when I encounter theorem 8.14 which comes almost right ...
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### Compute Riemann Sum

I was not formally taught how to evaluate Riemann sums using the summation rule, and so I am going off of solutions to other problems to apply to my own problem. However, I am stuck. Any help would ...
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### Theorem on Riemann integral of monotone functions

I want to prove the following: Theorem: Let $(f_n)$ be a sequence of monotone (integrable) functions $f_n : [0,\infty) \rightarrow \mathbb{R}$ (for $t_i \leq t_j$ we have $f_n(t_i) \leq f_n(t_j)$ such ...
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### Improper integral convergence implies the existence of an infinite series which its partial sum converges

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a Riemann-integrable function at $[0, \beta]$ for each $\beta \in (0, \infty)$. Suppose that $\forall x \in [0, \infty): \space f(x) \geq 0$. If the ...
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