# Questions tagged [riemann-sum]

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

188 questions
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### Understanding Stieltjes-Riemann

From my understanding from lectures, the Stieltjes-Riemann integral is a generalization of the Riemann integral. When using the identity function as integrator, the Riemann sum and Stieltjes-Riemann ...
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### Converting a series to a Riemann sum,

I am manipulating a series and have gotten this far: $$\lim_{n\to\infty} \sum_{m=1}^n \frac {(\frac{m}{n})^{p-1}}{1+ (\frac{m}{n})^p} \frac{1}{n}$$ I want to now say that this is a Riemann sum, ...
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### How to prove that : $\lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$

I am just having fun with this question: Is this true that $\displaystyle \lim_{n \rightarrow \infty}\frac{1}{n}\sum_1^n\frac{k^8}{(a+(k+b)^2)^4}=1$? I thought to change this to an integral, namely ...
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### How to show that a piecewise constant function is integrable, using the upper and lower sums?

Let $f(x) = \begin{cases} 1 &\mbox{if } 0\leq x<1 \\ 3 &\mbox{if } 1\leq x<2 \\ 2 &\mbox{if } 2\leq x\leq 3. \end{cases}$ Show that $f(x)$ is integrable by $(a)$ ...
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### Show $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b]

Let $f$ be bounded on a nondegenerate interval $[a,b]$. Prove that $f$ if integrable on $[a,b]$ if and only if $\epsilon > 0$, there exists a partition $P_{\epsilon}$ of [a,b] such that P is a ...
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### How does double Riemann sum actually work?

I'm in an advanced calculus class and studying double integral. My question is about how double Riemann sum actually work as algebraic steps? I mean I understand essentially it is summing up the ...
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### $\Delta x$ in the limit-definition, or Riemann-Sum-definition, of an Integral.

Every time I evaluate an integral as a Riemann Sum and I see $\Delta x$ as $\left(1/n\right)$, I think of an upper limit that could be $a+1$ and a lower limit that could be $a$, where $a$ is any real ...
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### Limit involving primes: $\lim_{n\to\infty}\frac{1}{n}\sum_{k = 1}^{n}\left(\frac{1}{2}-\frac{1}{4}\frac{\log \log p_k }{\log p_k} \right)$
Given function $f$ that is continuous and defined on the closed, finite inverval $[a,b]$ Also given any two partitions $P_1$ and $P_2$ and their common refinement $P$ which consists of all of the ...