Limit serie

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0 replies

jlacosta

Apr 2, 2025

0 replies

VJIMC 2015 Category I, Problem 1

ellipticfunction 9

N 3 minutes ago by Rohit-2006

Source: VJIMC2015

Problem 1
Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be differentiable on $\mathbb{R}$ . Prove that there exists $x \in [0, 1]$ such that
$\frac{4}{\pi} ( f(1) - f(0) ) = (1+x^2) f'(x) \ .$

9 replies

ellipticfunction

Aug 10, 2015

Rohit-2006

3 minutes ago

Difficult galois theory problem

dust_ 0

an hour ago

Source: Yau contest

Let $K = \mathbb{Q}(\sqrt{-3})$ , an imaginary quadratic field.
Does there exists a finite Galois extension $L/\mathbb{Q}$ which contains $K$ such that $Gal(L/\mathbb{Q})\cong Q$ ? Here $Q$ is the quaternion group with 8 elements $\{\pm1,\pm i,\pm j,\pm k\}$ , a finite subgroup of the group of units $\mathbb{H}^{\times}$ of the ring $\mathbb{H}$ of all Hamiltonian quaternions.

0 replies

dust_

an hour ago

0 replies

Matrix Row and column relation.

Schro 5

N an hour ago by Etkan

If ith row of a matrix A is dependent,Then ith column of A is also dependent and vice versa .

Am i correct...

5 replies

Schro

Yesterday at 2:54 PM

Etkan

an hour ago

D1023 : MVT 2.0

Dattier 0

5 hours ago

Source: les dattes à Dattier

Let $f \in C(\mathbb R)$ derivable on $\mathbb R$ with $\forall x \in \mathbb R,\forall h \geq 0, f(x)-3f(x+h)+3f(x+2h)-f(x+3h) \geq 0$
Is it true that $\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$

0 replies

Dattier

5 hours ago

0 replies

No more topics!

Limit serie

Moubinool 1

N Mar 30, 2025 by paxtonw

Source: Oral examination Ecole Polytechnique France

A(n) is a sequence given by
$A(n)=\frac{1}{n} \sum_{ k , integer, \sqrt{2}< k/n < \sqrt{2} +1} \frac{1}{\sqrt{k/n - \sqrt{2}}}$ Find limit of A(n) when n tend to +oo

1 reply

Moubinool

Mar 29, 2025

paxtonw

Mar 30, 2025

Limit serie

G H J

Reveal topic tags

calculus

G H BBookmark kLocked kLocked NReply

Source: Oral examination Ecole Polytechnique France

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Moubinool

5567 posts

Moubinool

#1 h Mar 29, 2025, 9:17 PM

Y by

A(n) is a sequence given by
$A(n)=\frac{1}{n} \sum_{ k , integer, \sqrt{2}< k/n < \sqrt{2} +1} \frac{1}{\sqrt{k/n - \sqrt{2}}}$ Find limit of A(n) when n tend to +oo

This post has been edited 6 times. Last edited by Moubinool, Mar 29, 2025, 10:09 PM

Z K Y

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paxtonw

28 posts

paxtonw

#2 h Mar 30, 2025, 12:54 AM • 1 Y

Y by MihaiT

A(n) = (1/n) × ∑ [1 / √(k/n − √2)],
where the sum is over integers k such that:
√2 < k/n < √2 + 1

As n --> ∞, this becomes a Riemann sum over the interval (√2, √2 + 1), so:

A(n) ≈ ∫ from √2 to (√2 + 1) of [1 / √(x − √2)] dx

Let u = x − √2, so the limits become 0 to 1:
∫ from 0 to 1 of (1 / √u) du = 2

limₙ --> ∞ A(n) = 2

Z K Y

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