Basically I understand the minimum is 7, if you think of the left one as sort of sticking upwards, and the right image is the image on the left rotated 90 degrees to the left, but like 3D (idk how to explain), and in the left image its basically the bumps from the right but you don't see them because you are looking at it directly from the top. Anyway, I know its C now, but I am not sure how I would solve to know if it is C or E.

Don't really know how to tag this but I was thinking of fractions and came across this identity (if you can call it that).

For any fraction a/b such that b ≠ 0 and a/b > -1, it can be represented as the sum from n = 1 to infinity of a / (a+b)ⁿ

I realized this by playing around with the infinite sum of a geometric progression where I imagined that any number can be the sum of an infinite geometric series, and that the series starts at it's common ratio

r / (1 - r) = a/b which is how I got a / (a+b)ⁿ

What I'm wondering now is that for the infinite sum, it is defined for |r| > 1, but in this version, a/b only follows the rules a/b > -1

Why is this and what property/formula have I stumbled upon?

The following example caught my attention-- this 2/7 should have been 3/7, right? Or could you please clarify it for me if I'm mistaken?

I'm now questioning if this book is reliable.

First semester in a physics degree, and I am struggling with an analysis class that is supposed to be the calculus equivalent. I simply don't know why some of these things work. Where do I learn techniques to solve problems like the one above and explanations as to why these technqiues work?

I have a friend who is in 10th grade. Although Taylor series are not yet covered in the high school curriculum, he learned derivatives, integrals, and Taylor series expansions by himself through YouTube and Wikipedia. He wrote these equations, but I don't know if they are correct.

l need to provide a counter example to this, I have made many attempts on a calculator and I cant figure it out

so what i do not understand is

1.what does he mean by we are no longer interested in the upper bound

1. how does this abomination |bn| ≥ δ > 0 help n' why did we use it n' how does it "lead" to 1/(|b||bn|)

2. why does bn need to be closer to b than to 0

4.from where did the (b)^2 come from

i do have some other Q but if i understand these first the rest should be fine

The resulting structure forms an hourglass-like shape with spirals and diagonal alignments. Can you spot any geometric patterns that could help predict the location of the next red point along these spiral tracks?

i know the squeeze theorem proof but it uses the area of the sector being x/2 which already assumes radians. in degrees the limit comes out as pi/180 and i can't see why changing units would change the limit so much

is it just that radians are the only unit where arc length equals the angle, so the derivative of sin comes out clean? feels like i'm missing something obvious

A general intuitive approach provided is that they are like what prime numbers are for composite numbers, but it still doesn't make sense, the relationship between finite simple groups and complex groups feels "more nested and yet sort of separate/disjoint for the lack of a better word", it doesn't fully capture how the group structure is assembled.

I understand the formal steps - take a maximal proper normal subgroup (N) of a group (G), form the quotient (G/N), and repeat the process with (N) until we reach the trivial subgroup. The successive quotients are then simple groups. But I still have a hard time intuitively grasping why these quotient groups (which are sets of cosets), are the building blocks of our original group, or how these quotient groups come together to form the original group?

Undergrad here with only an introductory level knowledge of group theory so excuse any ignorance.

Hello! I dont speak english so sorry if I explain something badly.
I'm having trouble with this, I'm not sure what is even the correct way to solve this, if I thought about it wrong!
The original equation on picture 2 and the right aswer is circled and has arrows pointing to it.

Proposal for a parametric iterated exponentiation operator for the simplification of exponential towers with a fixed base. In this project, I introduce a new mathematical formalism—denoted as $\text{j}_{m}^{n}(a)$—designed to overcome the analytical limitations of fixed-base power towers. The symbol "j" (representing the IPA phonetic symbol for a palatal fricative) serves as an operator that transforms a power tower into this specific notation. Here, $m$ is the parameter (the value to be raised), $n$ represents the power tower (indicating the number of times $m$ is raised), and $a$ is the base. My primary motivation for pursuing this innovation was the observation that no existing notation—to my knowledge—adequately addressed these structures; I sought to contribute in this area, as I have long been fascinated by power towers. Below are the properties I have been able to prove Properties:​1.-1.- Torre 1: \text{j}_{m}^{1}(a) con n siendo el 1, 2.- Torre 2: \text{j}_{m}^{2}(a) con n siendo 2, resultado: a^{m^m} 3.- Parámetro 0: \text{j}_{0}^{n}(a) con m siendo 0 (para n \ge 1), resultado: 1 4.- Parámetro 1: \text{j}_{1}^{n}(a) con m siendo 1, resultado: a 5.- Base unitaria: \text{j}_{m}^{n}(1) con a siendo 1, resultado: 1 6.- Distributiva de base: \text{j}_{m}^{n}(a \times b) = \text{j}_{m}^{n}(a) \times \text{j}_{m}^{n}(b) 7.- Parámetro mayor: Si m_1 < m_2, entonces \text{j}_{m_1}^{n}(a) < \text{j}_{m_2}^{n}(a) (para a > 1) 8.- Parámetro negativo: \text{j}_{-2}^{n}(a) = a^{-2} = \frac{1}{a^2} 9.- Torre fraccionaria: \text{j}_{m}^{0.5}(a) = a^{0.5} = \sqrt{a} 10.- Cociente de operadores: \frac{\text{j}_{m}^{n}(a)}{\text{j}_{m}^{k}(a)} = a^{\text{torre}(n) - \text{torre}(k)} 11.- Suma de operadores: \text{j}_{m}^{n}(a) + \text{j}_{m}^{k}(a) = \text{j}_{m}^{k}(a) \times \left(a^{\text{torre}(n) - \text{torre}(k)} + 1\right) And this concludes my presentation on my notation. Thank you for your attention and understanding. If I have made any errors, I welcome constructive criticism so that we may engage in a healthy and ongoing exchange of feedback. :)

I came across a question where assume ax^b+c = 3x^2+1, naturally many people directly write it as a= 3, b= 2 and c=1 just by comparing the LHS and RHS simultaneously. I believe many assume it to be a identity that is true for all R therefore you match the terms, but I feel like it's not mathematically rigourous

the above equation can be rearranged to ax^b+c-3x^2-1=0, assuming all a,b,c,x to be variables with their respective domains you get a unique solution for (a,b,c,x) that gives you a graph for all the possible values.

But when there's infinitely many solutions, whats the advantage in writing down a random pair by comparison. Would it be okay to just solve for x by taking random values for a,b and c?

Which viewpoint is more mathematically accurate? The identity one or that the above equation is a function and that you get infinite pairs of solutions and you can choose any one?

If f is an injective function f: X -> Y and y an element in the image of f, is the preimage of y through f a singleton subset of X or is it an element of X?

Edit: since I am not getting much replies on Magnet communities I am posting this here. Also the Ask Physics and Engineers don't allow image for some reason. If you could please help me with calculations or just estimations.

I am creating a board game where the pieces magnetise to strict locations on the board which is 2mm thick. If I make an array like shown in the first image and the magnets bellow and above the board are identical, what strength should the circular magnets be, given I choose a specific magnet for the edges (in this case a 20x3x3mm). I am guessing they should be a similar strength, but perhaps it is optimal to go with less or more force?

For the third image if I had those same block magnets, can I place some above the board so that they lock into a perpendicular orientation? or will they fight to align

What would be the pros and cons of using the axiom of dependent choice over the axiom of choice. Just the fact that with the axiom of dependent choice every set becomes measurable, but some vector spaces dont have a basis anymore. Is it even really worth it to make every set measurable?

Let’s say you have a ball that appears in a random position within a circle

Counting the position of the ball inside the circle as x, y, the initial fall angle as a, the speed of the ball as v

What would a function that explains the minimum number of balls to make a loop be like if the speed of the ball is always constant?

If Y is a subset of X, I may write X ⊇ Y and say "X contains Y". Sometimes, though, I wanna speak of whether or not something is an element of X.

Saying X contains x is wrong. Saying "X has x as an element" is right but a bit verbose for my liking. Is there a more direct way of saying this?

Hello, I am trying to wrap my head around this problem but don't seem to be getting the correct answer. The question is:

8 people are to stay in 2 rooms where a maximum of 6 people can stay in a room. In how many ways can the 8 people stay in 2 rooms?

The archetype of the mathematical prodigy is one of the most romanticized figures in human history. We marvel at Srinivasa Ramanujan arriving from South India with notebooks full of groundbreaking equations, Terence Tao scoring a 760 on the math SAT at age 8, and Newton rewriting physics in his twenties. For centuries, math was the ultimate arena of pure human intellect-a playground where a young, unpolluted mind could see patterns that adults missed.

But now, AI systems like DeepMind’s AlphaGeometry are solving Olympiad-level math problems and discovering brand-new mathematical insights on their own. This raises a massive question: Is the era of the young mathematical prodigy officially over?