Question

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k+\theta}{n})=\int_{0}^{1}f(x)dx$ where $\theta\in(0,1)$?

score 66 · Answer 1 · 2022-01-27 13:29:48

66

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$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k+\theta}{n})=\int_{0}^{1}f(x)dx$ where $\theta\in(0,1)$?

Published on 27 Jan 2022 - 13:29

score 135 · Answer 2 · 2022-01-29 06:25:05

135

Views

Conventionally $\int_2^0 f(x)dx:=-\int_0^2 f(x)dx$ whereas $\sum_{i=2}^0 := 0$ Why are definite integrals and series treated differently?

Published on 29 Jan 2022 - 6:25

score 164 · Answer 3 · 2026-03-27 14:57:33

164

Views

Convolution integral over non-uniform grids

Published on 27 Mar 2026 - 14:57

score 618 · Answer 4 · 2026-02-23 12:04:33

618

Views

When would we want to use uneven subintervals in a Riemann integral?

Published on 23 Feb 2026 - 12:04

score 128 · Answer 5 · 2026-03-30 15:53:31

128

Views

Change of variables for the Riemann sum?

Published on 30 Mar 2026 - 15:53

score 177 · Answer 6 · 2022-02-06 10:33:30

177

Views

Riemann sum simplifying for the integral of an exponential function

Published on 06 Feb 2022 - 10:33

score 49 · Answer 7 · 2022-02-24 01:27:13

49

Views

Find $\int_{-1}^{2}(5x^3+7x^2-9x+4)dx$ by Riemann sums

Published on 24 Feb 2022 - 1:27

score 2k · Answer 8 · 2022-02-27 08:42:49

2k

Views

Riemann sum approximation answer

Published on 27 Feb 2022 - 8:42

score 105 · Answer 9 · 2026-03-26 11:16:17

105

Views

Regarding Theorem 1.9, Conway, complex Analysis

Published on 26 Mar 2026 - 11:16

score 40 · Answer 10 · 2022-02-28 15:21:08

40

Views

Distance between two Riemann-integrable functions

Published on 28 Feb 2022 - 15:21

score 31 · Answer 11 · 2022-03-11 14:19:53

31

Views

Limit of partial Harmonic Sum as a bounded integral

Published on 11 Mar 2022 - 14:19

score 48 · Answer 12 · 2022-03-15 17:10:09

48

Views

Proof that a function is Riemann integrable by using specific hint.

Published on 15 Mar 2022 - 17:10

score 43 · Answer 13 · 2022-03-15 17:29:02

43

Views

Help with Proof of theorem about Riemann Integrability Criteria

Published on 15 Mar 2022 - 17:29

score 145 · Answer 14 · 2022-03-17 06:40:49

145

Views

How to take limits of 'almost Riemann' sums like $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} \cos (a \pi k \log(n)/n)$

Published on 17 Mar 2022 - 6:40

score 40 · Answer 15 · 2022-03-27 06:29:18

40

Views

Proof a piecewise function is darboux integrable on $[-1,3]$

Published on 27 Mar 2022 - 6:29

List Question

$\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(\frac{k+\theta}{n})=\int_{0}^{1}f(x)dx$ where $\theta\in(0,1)$?

Conventionally $\int_2^0 f(x)dx:=-\int_0^2 f(x)dx$ whereas $\sum_{i=2}^0 := 0$ Why are definite integrals and series treated differently?

Convolution integral over non-uniform grids

When would we want to use uneven subintervals in a Riemann integral?

Change of variables for the Riemann sum?

Riemann sum simplifying for the integral of an exponential function

Find $\int_{-1}^{2}(5x^3+7x^2-9x+4)dx$ by Riemann sums

Riemann sum approximation answer

Regarding Theorem 1.9, Conway, complex Analysis

Distance between two Riemann-integrable functions

Limit of partial Harmonic Sum as a bounded integral

Proof that a function is Riemann integrable by using specific hint.

Help with Proof of theorem about Riemann Integrability Criteria

How to take limits of 'almost Riemann' sums like $\lim_{n \to \infty} \sum_{k=0}^n \frac{1}{n} \cos (a \pi k \log(n)/n)$

Proof a piecewise function is darboux integrable on $[-1,3]$

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