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1500 questions
78
votes
5 answers
Help me solve my father's riddle and get my book back
My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty.
So today before leaving home he sneaked into my room and took the book I am currently…
user161516
78
votes
6 answers
Is there a slowest rate of divergence of a series?
$$f(n)=\sum_{i=1}^n\frac{1}{i}$$
diverges slower than
$$g(n)=\sum_{i=1}^n\frac{1}{\sqrt{i}}$$
, by which I mean $\lim_{n\rightarrow \infty}(g(n)-f(n))=\infty$. Similarly, $\ln(n)$ diverges as fast as $f(n)$, as $\lim_{n \rightarrow…
Meow
- 6,353
78
votes
4 answers
How would you discover Stokes's theorem?
Let $S$ be a smooth oriented surface in $\mathbb R^3$ with boundary $C$, and let $f: \mathbb R^3 \to \mathbb R^3$ be a continuously differentiable vector field on $\mathbb R^3$.
Stokes's theorem states that
$$
\int_C f \cdot dr = \int_S (\nabla…
littleO
- 51,938
78
votes
11 answers
Proof of triangle inequality
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
ivan
- 3,237
78
votes
9 answers
What is mathematical research like?
I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or…
TheHopefulActuary
- 4,784
78
votes
9 answers
What do $\pi$ and $e$ stand for in the normal distribution formula?
I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is:
$$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^2}{2\sigma^2}}},$$
However, what are $e$…
pimvdb
- 1,283
78
votes
5 answers
What am I doing when I separate the variables of a differential equation?
I see an equation like this:
$$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$
and solve it by "separating variables" like this:
$$y\textrm{d}y = e^x\textrm{d}x$$
$$\int y\textrm{d}y = \int e^x\textrm{d}x$$
$$y^2/2 = e^x + c$$
What am I doing when I solve…
Mark Eichenlaub
- 6,733
78
votes
3 answers
Intuitive explanation of a definition of the Fisher information
I'm studying statistics. When I read the textbook about Fisher Information, I couldn't understand why the Fisher Information is defined like this:
$$I(\theta)=E_\theta\left[-\frac{\partial^2 }{\partial \theta^2}\ln P(\theta;X)\right].$$
Could anyone…
maple
- 2,883
- 2
- 29
- 37
78
votes
2 answers
78
votes
8 answers
Multiple-choice: sum of primes below $1000$
I sat an exam 2 months ago and the question paper contains the problem:
Given that there are $168$ primes below $1000$. Then the sum of all primes
below 1000 is
(a) $11555$ (b) $76127$ (c) $57298$ (d) $81722$
My attempt to solve it: We know…
Sufaid Saleel
- 3,789
78
votes
9 answers
Is it bad form to write mysterious proofs without explaining what one intends to do?
Often when doing assignments, I find myself deliberately writing in a "mysterious" way. By this I mean that the reader usually will not understand what exactly is going on and what for, until the very end where all the things come together.
A…
Jaood
- 1,391
78
votes
5 answers
Drunk man with a set of keys.
I found this problem in a contest of years ago, but I'm not very good at probability, so I prefer to see how you do it:
A man gets drunk half of the days of a month. To open his house, he has a set of keys with $5$ keys that are all very similar,…
iam_agf
- 5,438
78
votes
3 answers
Why is $i! = 0.498015668 - 0.154949828i$?
While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!.
Curiously, Google's calculator dutifully informed me that $i!$ was, in fact, $0.498015668 - 0.154949828i$.
Why…
growse
- 803
78
votes
1 answer
How was the Monster's existence originally suspected?
I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.
For example, in ncatlab's article,
The Monster group was…
anon
- 151,657
78
votes
1 answer
Gerrymandering on a high-genus surface/can I use my powers for evil?
Somewhat in contrast to this question.
Let's say the Supreme Court has just issued a ruling that the upper and lower roads of an overpass need not be in the same congressional district. This makes states with lots of overpasses into high-genus…
Alex Becker
- 60,569