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1500 questions
65
votes
3 answers
What does "∈" mean?
I have started seeing the "∈" symbol in math. What exactly does it mean? I have tried googling it but google takes the symbol out of the search.
Locke
- 823
65
votes
6 answers
Midpoint-convexity and continuity implies convexity
Assume that function $f$ is continuous on an interval $(a,b)$ . Given that $$ f \left( \frac{x+y}{2} \right) \leqslant \frac{f(x) + f(y)}{2} \,,$$ how can I show that $f$ is convex?
Jack
- 863
65
votes
18 answers
What are some good ways to get children excited about math?
I'm talking in the range of 10-12 years old, but this question isn't limited to only that range. Do you have any advice on cool things to show kids that might spark their interest in spending more time with math? The difficulty for some to learn…
David McGraw
- 885
65
votes
17 answers
Interesting "real life" applications of serious theorems
As student in mathematics, one sometimes encounters exercises which ask you to solve a rather funny "real life problem", e.g. I recall an exercise on the Krein-Milman theorem which was something like: "You have a great circular pizza with $n$…
user149890
65
votes
5 answers
Subgroups of finitely generated groups are not necessarily finitely generated
I was wondering this today, and my algebra professor didn't know the answer. Are subgroups of finitely generated groups also finitely generated? I suppose it is necessarily true for finitely generated abelian groups, but is it true in general? And…
crasic
- 5,199
65
votes
6 answers
Is the power set of the natural numbers countable?
Some explanations: A set S is countable if there exists an injective function $f$ from $S$ to the natural numbers ($f:S \rightarrow \mathbb{N}$). $\{1,2,3,4\}, \mathbb{N},\mathbb{Z}, \mathbb{Q}$ are all countable. $\mathbb{R}$ is not countable. The…
Martin Thoma
- 10,247
65
votes
7 answers
Why does the volume of the unit sphere go to zero?
The volume of a $d$ dimensional hypersphere of radius $r$ is given by: $$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$ What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$. How can this be? For fixed…
probabilityislogic
- 1,055
65
votes
19 answers
How do I convince my students that the choice of variable of integration is irrelevant?
I will be TA this semester for the second course on Calculus, which contains the definite integral. I have thought this since the time I took this course, so how do I convince my students that for a definite integral $$\int_a^b f(x)\ dx=\int_a^b…
ireallydonknow
- 1,435
65
votes
7 answers
How to prove that a $3 \times 3$ Magic Square must have $5$ in its middle cell?
A Magic Square of order $n$ is an arrangement of $n^2$ numbers, usually distinct integers, in a square, such that the $n$ numbers in all rows, all columns, and both diagonals sum to the same constant. How to prove that a normal $3\times 3$ magic…
Archisman Panigrahi
- 2,736
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- 33
65
votes
2 answers
Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?
Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge? My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here.
Darius
- 1,374
- 11
- 17
65
votes
16 answers
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that $$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction. Thanks
Steve
- 419
65
votes
6 answers
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq n} \frac{1}{k}$, and failed. But should we…
Srivatsan
- 26,871
65
votes
1 answer
Is the largest root of a random polynomial more likely to be real than complex?
Posted on MO since it is unanswered in MSE It is known that the number of real roots of a random polynomial with real coefficients is much smaller than the number of complex roots. WLOG, assume that the coefficients are uniformly random in $(-1,1)$…
Nilotpal Sinha
- 24,874
65
votes
1 answer
Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion
I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship. In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean space (I'm talking about isometry of an open set -…
Selena Ballerina
- 2,775
65
votes
4 answers
Are there open problems in Linear Algebra?
I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory. There are a lot of open problems and conjectures in…
Mebat
- 1,331