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1500 questions
71
votes
8 answers
Dominoes and Induction, or How Does Induction Work?
I've never really understood how math induction is supposed to work. You have these 3 steps: Prove true for base case ( $n=0$ or $n=1$ or whatever) Assume true for $n=k$ . Call this the induction hypothesis . Prove true for $n=k+1$ , somewhere using…
bobobobo
- 9,822
71
votes
1 answer
Baby/Papa/Mama/Big Rudin
Recently, I was looking for the reviews of some Analysis books while encountered terms such as Baby/Papa/Mama/Big Rudin. Firstly, I thought that these are the names of a book! But it turned out that these are some nick names used for the books of…
Hosein Rahnama
- 15,676
71
votes
9 answers
Why is $\pi $ equal to $3.14159...$?
Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any circle by its diameter, you get the same number, and…
gphilip
- 837
71
votes
2 answers
Period of the sum/product of two functions
Suppose that period of $f(x)=T$ and period of $g(x)=S$, I am interested what is a period of $f(x) g(x)$? period of $f(x)+g(x)$? What I have tried is to search in internet, and found following link for this. Also I know that period of $\sin(x)$ is…
71
votes
13 answers
What is exactly the difference between a definition and an axiom?
I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall n:0\neq S(n)$, or in English, that zero isn't the…
wythagoras
- 25,856
71
votes
1 answer
Is this determinant identity known?
Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the main diagonal all equal -1 and entries on even lower…
Vincent
- 11,535
70
votes
1 answer
Evaluating sums and integrals using Taylor's Theorem
Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt $$ We can use this to evaluate integrals. For example, consider $f(x)=\frac{b!x^{b+n+1}}{(b+n+1)!}$. This has…
Pauly B
- 5,352
70
votes
4 answers
Does the string of prime numbers contain all natural numbers?
Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
Buddha
- 1,266
- 11
- 16
70
votes
7 answers
What is the oldest open problem in geometry?
Geometry is one of the oldest branches of mathematics, and many famous problems have been proposed and solved in its long history. What I would like to know is: What is the oldest open problem in geometry? Also (soft questions) : Why is it so hard?…
Sudoku Polo
- 693
70
votes
5 answers
Does every prime divide some Fibonacci number?
I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can I do this? Originally, I was trying to show that…
Chanhee Jeong
- 1,048
70
votes
3 answers
Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?
I have a couple of questions about tensor products: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a multilinear map $V^m \to V^{\otimes n}$? What is the…
Eric Auld
- 29,287
70
votes
4 answers
Difference between NFA and DFA
In very simple terms please, all resources I'm finding are talking about tuples and stuff and I just need a simple explanation that I can remember easily because I keep getting them mixed up.
Ogen
- 2,335
70
votes
3 answers
Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
One way to evaluate $ \displaystyle\int_{0}^{z} \log \Gamma(x) \, \mathrm dx $ is in terms of the Barnes G-function . $$ \int_{0}^{z} \log \Gamma(x) \, \mathrm dx = \frac{z}{2} \log (2 \pi) + \frac{z(1-z)}{2} + z \log \Gamma(z) - \log G(z+1)$$…
Random Variable
- 45,361
70
votes
27 answers
What are some conjectures of your own?
Background: Although this site is most-often used for specific one-off questions, many of the highest scored questions (also on MathOverflow ), which gather a lot of attention to the site are about informal lists. So, in the theme of, but in…
Graviton
- 4,688
70
votes
2 answers
Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?
This question is inspired by my answer to the question How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$? . The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer $k$ ) came up, and I noticed that $f(1) = e-1$ was…
marty cohen
- 110,916