D1023 : MVT 2.0

by Dattier, Apr 29, 2025, 1:04 PM

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 $\forall (a,b) \in\mathbb R^2, |f(a)-f(b)|\leq \max\left(\left|f'\left(\dfrac{a+b} 2\right)\right|,\dfrac {|f'(a)+f'(b)|}{2}\right)\times |a-b|$

real analysis

Limit of the sum of a symmetric polynomial evaluated at sequence terms

by tom-nowy, Apr 29, 2025, 1:27 AM

The real sequence $\{ a_n \}_{n=1,2,3,\ldots}$ satisfies $0 \le a_n \le 1 \, (\forall n \in \mathbb{N} )$ and $\lim_{n\to\infty} a_n = 1$ . Find
$\lim_{n\to\infty} n \left( \;1 \; +\; \sum_{k=1}^n \frac{ (-1)^k}{k+1} \sum_{1 \le j_1 \le j_2 \le \cdots \le j_k \le n} a_{j_1} a_{j_2} \cdots a_{j_k} \right).$

Sequences

Limit of sequence

limit of sum

Probability

by Qebehsenuef, Apr 28, 2025, 11:45 PM

A mouse initially occupies cage A and is trained to change cages by going through a tunnel whenever an alarm sounds. Each time the alarm sounds, the mouse chooses any of the tunnels adjacent to its cage with equal probability and without being affected by previous choices. What is the probability that after the alarm sounds 23 times the mouse occupies cage B?

Attachments:

probability

probability and stats

Matrix Row and column relation.

by Schro, Apr 28, 2025, 2:54 PM

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

Am i correct...

linear algebra

UC Berkeley Integration Bee 2025 Qualifying Exam

by Silver08, Apr 27, 2025, 1:47 AM

Good luck and have fun!!!

1. $\int_{-\sqrt[3]{2}}^{1}(x^3+x^6)(x^2+2x^5)dx$
2. $\int \frac{e^x}{1+e^{2x}}+\frac{x}{1+x^2}dx$
3. $\int_0^\pi x^4\cos(x)dx$
4. $\int_{0}^{\frac{\pi}{2}}\sin^2(x)\cos^2(x)dx$
5. $\int_1^2\frac{\sqrt{x^2+2\sqrt{x^2-1}}}{\sqrt{x^2-1}}dx$
6. $\int_2^4\frac{(2-x)}{(x-1)(x-5)}dx$
7. $\int_0^\infty \frac{dx}{x^2-2x+2}$
8. $\int (e^x+\ln(x))\left(e^x+\frac{1}{x}\right)dx$
9. $\int_{-\infty}^{\infty}\frac{\ln(x^2+1)}{x^2}dx$
10. $\int_0^1 \frac{dx}{\sqrt{1+\sqrt{x}}-\sqrt{1-\sqrt{x}}}$
11. $\int_0^1 2^{\ln(x)}dx$
12. $\int \frac{d}{dx}\left [ \frac{e^x}{\ln(x)} \right] \cdot \frac{\ln(x)}{e^x} dx$
13. $\int_1^\infty \ln(x)-\frac{1}{2}\ln(x^2+1)dx$
14. $\int_0^1 2x\sqrt{x-x^2}dx$
15. $\int_0^1 \ln^3(x)dx$
16. $\int_0^\pi\frac{\sqrt{\csc(x)-\sin(x)}}{\sqrt{\sin(x)}+1}dx$
17. $\int_0^6\frac{x}{\sqrt{x+\sqrt{x+\sqrt{x+...}}}}dx$
18. $\int_0^\infty \frac{\sin(x)\cos(2x)\cos^2(x)}{x}dx$
19. $\int_0^{\infty}\frac{dx}{1+x+x^2+x^3}$
20. $\int_{-\infty}^{\infty}\frac{(x^5-x^3)}{(x^2-1)^4+x^4}dx$

college

calculus

integration

D1021 : Does this series converge?

by Dattier, Apr 26, 2025, 4:29 PM

Is this series $\sum \limits_{k\geq 1} \dfrac{\ln(1+\sin(k))} k$ converge?

real analysis

series

Triple Sum

by P162008, Apr 26, 2025, 9:37 AM

Evaluate $\Omega = \lim_{n \to \infty} \frac{1}{n^5} \sum_{i=1}^{n} \sum_{k=1}^{2n} \sum_{k=1}^{3n} \frac{i(jk + 6n^2 - 3jn - 2kn) - jkn + n^2(3j + 2k) - 6n^3}{\sqrt{i^2 + j^2 + k^2 - i(a - 2n) + j(b - 4n) + k(c - 6n) + c + 14n^2}}$

This post has been edited 1 time. Last edited by P162008, Apr 26, 2025, 9:38 AM
Reason: Typo

calculus

Polynomial Limit

by P162008, Apr 25, 2025, 1:47 AM

Let $p = \lim_{y\to\infty} \left(\frac{2}{y^2} \left(\lim_{z\to\infty} \frac{1}{z^4} \left(\lim_{x\to 0} \frac{((y^2 + y + 1)x^k + 1)^{z^2 + z + 1} - ((z^2 + z + 1)x^k + 1)^{y^2 + y + 1}}{x^{2k}}\right)\right)\right)^y$ where $k \in N$ and $q = \lim_{n\to\infty} \left(\frac{\binom{2n}{n}. n!}{n^n}\right)^{1/n}$ where $n \in N$ . Find the value of $p.q.$

This post has been edited 2 times. Last edited by P162008, Monday at 6:32 AM
Reason: Typo

A Ball-Drawing problem

by Vivacious_Owl, Apr 24, 2025, 2:58 AM

There are N identical black balls in a bag. I randomly take one ball out of the bag. If it is a black ball, I throw it away and put a white ball back into the bag instead. If it is a white ball, I simply throw it away and do not put anything back into the bag. The probability of getting any ball is the same.
Questions:
1. How many times will I need to reach into the bag to empty it?
2. What is the ratio of the expected maximum number of white balls in the bag to N in the limit as N goes to infinity?

probability and stats

Time required for draining out water

by Kunihiko_Chikaya, Mar 14, 2006, 6:23 AM

Let $H>0,\ R>0$ and $O$ be the origin, $P\ (R,\ 0\ ,H).$ In space given the vessel formed by the revolution of the segment $OP$ about $z$ -axis. Denote the hight from $O$ to the surface of water by $h.$ When the vessel is filled with water, drain the water out such that the displacement per time is $\sqrt{h}$ at the time of $h,$ that is to say, if the total volume of the water drained out till the time $t$ is $V(t),$ then $\frac{dV}{dt}=\sqrt{h}$ holds. Find the time required for draining out water.

calculus

calculus computations

Submit

I still exist as well.
by G.G.Otto, Aug 11, 2023, 2:44 AM
hello I'm still here lol
by player01, Aug 6, 2022, 6:24 PM
[REVIVAL] I will start posting more calculator relating posts very soon. Even though school has been busy, I have been programming my calculators a decent amount, so I have a lot to share...
by sonone, Feb 18, 2022, 10:29 PM
wow its been like 2.5 years since geo class
by pieMax2713, Feb 4, 2022, 8:38 PM
@violin21, I've been very busy with school lately and haven't been able to add another lesson. I will when i get a free moment
by sonone, Aug 19, 2021, 12:45 AM
ORZ CODER
by samrocksnature, Aug 9, 2021, 9:57 PM
Could you make more Asymptote lessons on your "How to do Asymptote" blog?
by violin21, Aug 9, 2021, 7:26 PM
You can take it, just C&P the CSS into your CSS area
by sonone, Apr 17, 2021, 10:08 PM
how can we take the CSS if we have permission to not take it?
by GoogleNebula, Apr 17, 2021, 5:22 PM
That is awesome!
by sonone, Apr 15, 2021, 10:09 PM
I modified your dodecahedron and got:
[asy]
import three;
import solids;
size(300);
currentprojection=orthographic(0,1.3,1.2);
light(0,5,10);

real phi=(sqrt(6)+1)/3;
real g=(phi-1)/2;
real s=1/2;
real a=sqrt(1-phi*phi/4-g*g)+phi/2;

triple[] d;
d[0]=(phi
by Andrew2019, Mar 26, 2021, 12:15 AM
Not too many, just changing the color here and there. I really like your CSS!
by sonone, Feb 2, 2021, 10:35 AM
Nice!

I see you're making changes to the CSS.
by G.G.Otto, Feb 1, 2021, 9:26 PM
I'm learning Java now!
by sonone, Feb 1, 2021, 5:56 PM
And I took part of it from CaptainFlint and then added a ton of modifications.
by G.G.Otto, Dec 1, 2020, 8:56 AM
fixed $\text{[math]}$
by sonone, Apr 25, 2020, 8:43 PM
Change in progress...
by sonone, Apr 25, 2020, 8:42 PM
When it says "At least 11 users clicked that button" it's incorrect since one person could have just posted 11 times. It should say "At most 11 users click that button".
by G.G.Otto, Apr 25, 2020, 8:26 PM
see this PM
by sonone, Apr 25, 2020, 7:03 PM
I can't get the cursor to work.
by G.G.Otto, Apr 25, 2020, 6:38 PM
If the cursor is confusing, where you are actually pointing in a front of the ship
by sonone, Apr 25, 2020, 11:04 AM
It's ok $\text{[math]}$
by sonone, Apr 23, 2020, 5:30 PM
Sorry I haven't been posting. I've got a Research Paper (for Roman History) due in a week.
by G.G.Otto, Apr 22, 2020, 11:15 PM
You should download it, as it's really useful when learning python. You can find out more here
by G.G.Otto, Apr 19, 2020, 10:04 PM
No. I don't have IDLE.
by sonone, Apr 18, 2020, 7:49 PM
Do you have IDLE installed on your computer? If so, I can give you a couple really cool codes that made that don't work in the AoPS widgets.
by G.G.Otto, Apr 14, 2020, 6:47 AM
I don't know how to change it....
by G.G.Otto, Apr 5, 2020, 5:58 PM
Do you know how the change the into thing above my avatar a different color? It's too hard to see.
by sonone, Apr 5, 2020, 2:08 PM
WELCOME TO MY BLOG!
by sonone, Mar 31, 2020, 4:56 PM

98 shouts

sonone's place

by Dattier, Apr 29, 2025, 1:04 PM

by tom-nowy, Apr 29, 2025, 1:27 AM

by Qebehsenuef, Apr 28, 2025, 11:45 PM

by Schro, Apr 28, 2025, 2:54 PM

by Silver08, Apr 27, 2025, 1:47 AM

by Dattier, Apr 26, 2025, 4:29 PM

by P162008, Apr 26, 2025, 9:37 AM

by P162008, Apr 25, 2025, 1:47 AM

by Vivacious_Owl, Apr 24, 2025, 2:58 AM

by Kunihiko_Chikaya, Mar 14, 2006, 6:23 AM