Question

Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable

score 58 · Answer 1 · 2023-09-03 14:53:08

58

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Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable

Published on 03 Sep 2023 - 14:53

score 47 · Answer 2 · 2023-09-06 19:38:51

47

Views

Area under a curve using the rectangle method

Published on 06 Sep 2023 - 19:38

score 44 · Answer 3 · 2023-09-30 19:04:01

44

Views

Theorem on Riemann integral of monotone functions

Published on 30 Sep 2023 - 19:04

score 84 · Answer 4 · 2026-03-26 01:06:08

84

Views

Infimum of the Upper Sums

Published on 26 Mar 2026 - 1:06

score 107 · Answer 5 · 2023-10-11 13:14:47

107

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Calculate $\int_0^1 x^2~dx$ only by using upper-/lower Darboux sums

Published on 11 Oct 2023 - 13:14

score 60 · Answer 6 · 2023-10-14 23:25:02

60

Views

Telescopic Riemann sum

Published on 14 Oct 2023 - 23:25

score 44 · Answer 7 · 2026-03-26 01:02:42

44

Views

Help for Telescopic Riemann sum

Published on 26 Mar 2026 - 1:02

score 87 · Answer 8 · 2023-10-15 19:10:29

87

Views

Evaluating $\lim_{n\to\infty}\sum_{k=0}^{n/2-1} \frac{k}{(n-k)^2} $

Published on 15 Oct 2023 - 19:10

score 55 · Answer 9 · 2026-03-27 01:46:39

55

Views

A question that was related to the evaluation about the errors between Riemann sum and double integrals

Published on 27 Mar 2026 - 1:46

score 26 · Answer 10 · 2026-02-23 06:42:26

26

Views

A question on exercise 10.6(b) in Munkres - Analysis on Manifolds

Published on 23 Feb 2026 - 6:42

score 58 · Answer 11 · 2026-03-25 23:42:38

58

Views

Definite integral of ln(x) from 1 to a as the limit of a Riemann sum

Published on 25 Mar 2026 - 23:42

score 60 · Answer 12 · 2023-11-03 20:31:43

60

Views

Where does the differential come from when passing from a discrete sum to a continuous case, for instance for the case of the inner product?

Published on 03 Nov 2023 - 20:31

score 33 · Answer 13 · 2023-11-29 17:45:48

33

Views

Combination of function values on subinterval is bounded by $\inf$/$\sup$ of function.

Published on 29 Nov 2023 - 17:45

score 48 · Answer 14 · 2023-11-30 19:37:48

48

Views

Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$

Published on 30 Nov 2023 - 19:37

score 82 · Answer 15 · 2023-12-06 03:42:52

82

Views

Find a definite integral which represent $ \lim_{n\to\infty}{\frac{1}{n}\sum_{k=1}^{n}{\sqrt\frac{k}{n+k}}} $

Published on 06 Dec 2023 - 3:42

List Question

Let $f : [0,1] \to \mathbb R$ satisfy $\sum_1^n |f(t_i) - f(t_{i-1})|^2 < 100$ for any $ \leq t_0 < t_1 < ...< t_n \leq 1$. Prove $f$ is integrable

Area under a curve using the rectangle method

Theorem on Riemann integral of monotone functions

Infimum of the Upper Sums

Calculate $\int_0^1 x^2~dx$ only by using upper-/lower Darboux sums

Telescopic Riemann sum

Help for Telescopic Riemann sum

Evaluating $\lim_{n\to\infty}\sum_{k=0}^{n/2-1} \frac{k}{(n-k)^2} $

A question that was related to the evaluation about the errors between Riemann sum and double integrals

A question on exercise 10.6(b) in Munkres - Analysis on Manifolds

Definite integral of ln(x) from 1 to a as the limit of a Riemann sum

Where does the differential come from when passing from a discrete sum to a continuous case, for instance for the case of the inner product?

Combination of function values on subinterval is bounded by $\inf$/$\sup$ of function.

Suppose that for each $a < b$ s.t $a, b \in [0, 1]$ there exists $t_1, t_2 \in [a, b]$ which satisfies $g(t_2) \leq f(t_1)$

Find a definite integral which represent $ \lim_{n\to\infty}{\frac{1}{n}\sum_{k=1}^{n}{\sqrt\frac{k}{n+k}}} $

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