User Yiorgos S. Smyrlis

39 votes

1 answer

3k views

If $\,\lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}\,$ exists for every $x$, what does this imply for $f$?

asked Mar 22, 2018 at 22:05

34 votes

3 answers

1k views

Convergence of a series with repeated sines

calculus sequences-and-series analysis convergence-divergence

asked Dec 8, 2013 at 15:29

29 votes

0 answers

539 views

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

real-analysis sequences-and-series uniform-distribution

asked Jan 26, 2018 at 7:17

24 votes

3 answers

5k views

$AB=BA$ implies $AB^T=B^TA$ when $A$ is normal

linear-algebra matrices matrix-equations matrix-calculus

asked Feb 13, 2014 at 16:19

22 votes

6 answers

4k views

A problem V.I. Arnold solved as a primary school student

algebra-precalculus soft-question education math-history popular-math

asked Dec 31, 2013 at 0:04

19 votes

2 answers

699 views

Show that $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

calculus integration analysis definite-integrals residue-calculus

asked Jan 16, 2014 at 15:38

18 votes

2 answers

716 views

Finding simply connected open sets between compact ones and general open ones in $\mathbb R^2$.

real-analysis general-topology connectedness

asked Feb 10, 2015 at 11:04

15 votes

2 answers

257 views

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

calculus sequences-and-series convergence-divergence summation logarithms

asked Mar 28, 2018 at 10:56

15 votes

1 answer

594 views

Is it true that $\,(1+\sin n)^{1/n}\to 1$?

calculus real-analysis sequences-and-series

asked Jan 18, 2017 at 17:08

13 votes

3 answers

599 views

A problem with 26 distinct positive integers

combinatorics graph-theory recreational-mathematics contest-math

asked Dec 26, 2013 at 20:26

12 votes

1 answer

521 views

Integer parts of multiples of irrationals

number-theory analysis irrational-numbers ceiling-and-floor-functions set-partition

asked Dec 11, 2013 at 20:20

11 votes

1 answer

2k views

Solutions of autonomous ODEs are monotonic

calculus real-analysis analysis ordinary-differential-equations

asked May 13, 2015 at 19:19

10 votes

1 answer

278 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

calculus analysis recreational-mathematics contest-math functional-equations

asked Dec 20, 2013 at 17:14

10 votes

1 answer

371 views

If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing

real-analysis calculus derivatives continuity monotone-functions

asked Jul 13, 2019 at 15:47

9 votes

0 answers

192 views

Does the series $\sum \frac{\sin p}{p}$, where $p$ prime, converge? [duplicate]

calculus sequences-and-series convergence-divergence prime-numbers

asked Jun 17, 2018 at 14:38

9 votes

3 answers

2k views

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

analysis functional-analysis fourier-analysis sobolev-spaces banach-algebras

asked Jun 5, 2014 at 14:41

9 votes

1 answer

322 views

The grey area is equal to the white area

euclidean-geometry recreational-mathematics

asked Jun 16, 2016 at 12:24

8 votes

2 answers

522 views

Existence of a simple closed curve which is not null-homotopic

algebraic-topology

asked Oct 5, 2015 at 18:25

8 votes

1 answer

183 views

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

calculus real-analysis analysis contest-math

asked Dec 28, 2013 at 22:54

8 votes

4 answers

629 views

Area of a hexagon from the distances of the opposite sides

trigonometry euclidean-geometry

asked Jun 26, 2017 at 14:29

7 votes

1 answer

608 views

Lebesgue measurable but not Riemann integrable

lebesgue-integral riemann-integration

asked Mar 19, 2019 at 13:14

7 votes

1 answer

236 views

For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square?

number-theory elementary-number-theory diophantine-equations

asked Jul 21, 2016 at 16:24

6 votes

1 answer

222 views

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

real-analysis continuity irrational-numbers rational-numbers baire-category

asked Nov 12, 2015 at 11:37

6 votes

1 answer

94 views

Uniqueness is a consequence of a non-vanishing autonomous flux

calculus real-analysis ordinary-differential-equations

asked Dec 20, 2013 at 19:45

5 votes

1 answer

783 views

Functional equations leading to sine and cosine

calculus real-analysis trigonometry continuity functional-equations

asked Jan 17, 2014 at 8:49

5 votes

3 answers

196 views

Semigroup implies exponential

linear-algebra ordinary-differential-equations matrix-equations

asked Dec 16, 2013 at 22:02

5 votes

3 answers

147 views

What is the value of the limit $\lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}}$?

calculus sequences-and-series limits

asked Aug 21, 2014 at 11:21

5 votes

1 answer

179 views

Characterisation of convergence of bounded sequences via ultra-filters

real-analysis general-topology convergence-divergence filters nonstandard-analysis

asked Jan 18, 2016 at 12:43

4 votes

1 answer

400 views

Odd and even square roots of $z^2-1$

complex-analysis analysis analyticity radicals

asked Dec 14, 2013 at 23:54

4 votes

0 answers

115 views

Proof of the Three Utilities Problem with Complex Analysis?

complex-analysis graph-theory reference-request

asked Nov 16, 2019 at 8:55

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50 Questions

If $\,\lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}\,$ exists for every $x$, what does this imply for $f$?

Convergence of a series with repeated sines

The sequence $\,a_n=\lfloor \mathrm{e}^n\rfloor$ contains infinitely many odd and infinitely many even terms

$AB=BA$ implies $AB^T=B^TA$ when $A$ is normal

A problem V.I. Arnold solved as a primary school student

Show that $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

Finding simply connected open sets between compact ones and general open ones in $\mathbb R^2$.

If $\,a_n\searrow 0\,$ and $\,\sum_{n=1}^\infty a_n<\infty,\,$ does this imply that $\,n\log n\, a_n\to 0$?

Is it true that $\,(1+\sin n)^{1/n}\to 1$?

A problem with 26 distinct positive integers

Integer parts of multiples of irrationals

Solutions of autonomous ODEs are monotonic

A functional relation which is satisfied by $\cos x$ and $\sin x$

If $f$ is continuous and $f'(x)\ge 0$, outside of a countable set, then $f$ is increasing

Does the series $\sum \frac{\sin p}{p}$, where $p$ prime, converge? [duplicate]

For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

The grey area is equal to the white area

Existence of a simple closed curve which is not null-homotopic

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

Area of a hexagon from the distances of the opposite sides

Lebesgue measurable but not Riemann integrable

For which values of $n$ the sum $\sum_{k=1}^n k^2$ is a perfect square?

Is there a function, continuous on the irrationals, with rational values, nowhere locally constant?

Uniqueness is a consequence of a non-vanishing autonomous flux

Functional equations leading to sine and cosine

Semigroup implies exponential

What is the value of the limit $\lim_{a\searrow 0}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^{a}}$?

Characterisation of convergence of bounded sequences via ultra-filters

Odd and even square roots of $z^2-1$

Proof of the Three Utilities Problem with Complex Analysis?