Questions tagged [faq]
This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.
125 questions
4 votes
1 answer
322 views
Strategies for Evaluating $\int\frac{\mathrm dx}{(ax^2+bx+c)\sqrt{px^2+qx+r}}$, $a,b,c,p,q,r\in\mathbb R$ and $ap\neq0$
Question: What strategy/ies can be used for evaluating integrals of the type $\displaystyle\int\frac{\mathrm dx}{(ax^2+bx+c)\sqrt{px^2+qx+r}}$ where $a,b,c,p,q,r$ are real numbers and neither of $a$ ...
- 9,021
12 votes
1 answer
433 views
What is a good textbook to learn about infinite-dimensional manifolds?
The goal of this question is to serve as an "abstract duplicate target", as there are currently many different questions on this site about this exact topic. Which textbooks are there to ...
- 5,080
3 votes
2 answers
849 views
Overview of basic results in Stochastic Calculus
Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus? These can be in the form of resources (preferably accessible online) as well as directly writing out ...
- 4,927
0 votes
1 answer
788 views
Interpreting the row echelon form of an (augmented) matrix
Once we perform row reduction on a matrix, to put it in row echelon form, what does this tell us? How should we interpret the results? This question is intended as an FAQ. While plenty of questions ...
- 53.8k
37 votes
3 answers
2k views
Are there infinitely many primes of the form [X]? We probably don't know.
Are there infinitely many primes of the form [expression]? (We probably don't know. Sorry.) This question appears pretty often, with any number of various expressions. The sad reality is that the ...
- 3,167
10 votes
2 answers
2k views
Is there any variation known to the sum of two squares theorem?
Originally posed by Fermat and subsequently generalized as sum of two squares theorem, we can see the following statement. An integer greater than one can be written as a sum of two squares if and ...
- 382
5 votes
1 answer
2k views
How to solve linear recurrence relations with constant coefficients.
As questions regarding sequences that verifies a linear recurrence relation with constant coefficients are posted very often on this site and that there appear to be no reference post about it, so I ...
- 30k
0 votes
2 answers
5k views
Examples and Counterexamples of Relations which Satisfy Certain Properties
Definition: Given a set $X$, a relation $R$ on $X$ is any subset of $X\times X$. A relation $R$ on $X$ is said to be reflexive if $(x,x) \in R$ for all $x \in X$, irreflexive if $(x,x) \not\in R$ ...
- 33k
0 votes
5 answers
7k views
Proving that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$
How can one prove that among any $2n - 1$ integers, there's always a subset of $n$ which sum to a multiple of $n$? It is not hard to see this is equivalent to show that among $2n-1$ residue classes ...
- 53.4k
8 votes
3 answers
2k views
How to solve homogeneous linear recurrence relations with constant coefficients?
Consider a sequence $(a_n)_{n\in\mathbb N}$ defined by $k$ initial values $(a_1,\dots,a_k)$ and $$a_{n+k}=c_{k-1}a_{n+k-1}+\dots+c_0a_n$$ for all $n\in\mathbb N$. What are some ways to get closed ...
2 votes
0 answers
142 views
Proof of reduced upper-tail inequality for standard normal distribution [duplicate]
X∼N(0,1), then to prove that for x>0, $$ P(X>x)≤ \frac{1}{2}exp(−x^2/2) $$ I know how to prove the other two kinds of upper-tail inequality for standard normal distribution like this one $$exp(−x^...
- 23
5 votes
1 answer
215 views
When does $p^2$ divide $an^k + bp$?
In the ongoing effort of dealing with abstract duplicates. This question is about the lemma: Lemma Let $k \ge 2$, $p$ prime and $a$ coprime to $p$. Then $$p^2\!\mid a n^k+ bp\iff p\mid n,b.$$ This ...
- 28.1k
26 votes
1 answer
7k views
Getting different answers when integrating using different techniques
Question: Is it possible to get multiple correct results when evaluating an indefinite integral? If I use two different techniques to evaluate an integral, and I get two different answers, have I ...
- 33k
3 votes
6 answers
1k views
What is the error in this fake proof which uses series to show that $1=0$?
A common "trick" for obtaining a closed form of a geometric series is to define $$ R := \sum_{k=0}^{\infty} r^k, $$ then manipulate the series as follows: \begin{align} R - rR &= \sum_{k=0}^{\...
- 33k
33 votes
6 answers
8k views
How can a function with a hole (removable discontinuity) equal a function with no hole?
I've done some research, and I'm hoping someone can check me. My question was this: Assume I have the function $f(x) = \frac{(x-3)(x+2)}{(x-3)}$, so it has removable discontinuity at $x = 3$. We ...
- 2,589