## Top new questions this week:

### Explain "homotopy" to me

I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately. Please, an answer I need is ...

(general-topology) (algebraic-topology)

### Can an infinite sum of irrational numbers be rational?

Let $S = \sum_ {k=1}^\infty a_k$ where each $a_k$ is positive and irrational. Is it possible for $S$ to be rational, considering the additional restriction that non of the $a_k$'s is a linear ...

(real-analysis) (abstract-algebra) (summation)

### Proving the existence of a proof without actually giving a proof

In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. ...

(logic) (reference-request) (proof-theory)

### $-1$ as the only negative prime.

I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than $1$. This got me thinking about negative prime numbers though, and I soon realized that, for ...

(number-theory) (prime-numbers)

### Which derivatives are eventually periodic?

What derivatives are eventually periodic? I have noticed that is $a_{n}=f^{(n)}(x)$, the sequence $a_{n}$ becomes eventually periodic for a multitude of $f(x)$. If $f(x)$ was a polynomial, and ...

(calculus) (derivatives)

### Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...

(calculus) (differential-equations) (derivatives)

### What are some surprising appearances of $e$?

I recently came across the following beautiful and seemingly out-of-the-blue appearance of $e$: $E[\xi]=e$, where $\xi$ is a random variable that is defined as follows. It's the minimum number of ...

(soft-question) (education) (big-list)

## Greatest hits from previous weeks:

### Quadratic equation - Alpha and Beta Roots

If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ...

### Partial derivative in gradient descent for two variables

I've started taking an online machine learning class, and the first learning algorithm that we are going to be using is a form of linear regression using gradient descent. I don't have much of a ...

(calculus)

### A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...

(sequences-and-series) (convergence) (summation) (power-series)

### Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$X_m = \{A \in GL_n( \bf F_p): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear ...

(linear-algebra) (matrices) (finite-groups) (field-theory)
I have a machine. It has two states, broken or working. If it is working, then it will be broken with probability $q=0.1$. If the machine is working, I will make \\$1000 dollar a day. If it is broken, ...