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MathFail
  • Member for 2 years, 7 months
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26 votes
2 answers
1k views

A Tough Series: $\sum_{n=0}^\infty2^na_{n+1}\sqrt{a_n}=\frac{\Gamma^2(\frac14)-4\cdot\Gamma^2(\frac34)}{8\sqrt{2\pi}}$

16 votes
2 answers
720 views

Sophomore's Eternal Dream $\int_0^1\cdots\int_0^1 (t_1t_2\cdots t_n)^{t_1t_2\cdots t_n} dt_1 dt_2\cdots dt_n$

11 votes
1 answer
655 views

Tough integral: $\int_0^1 \frac{\arctan(x^{3+\sqrt{8}})}{1+x^2}dx$

11 votes
0 answers
235 views

Proof check: If $G$ is an infinite group, then it has infinite number of subgroups. [duplicate]

10 votes
5 answers
493 views

Why do we choose only the positive sign when substituting $x=\sin(t)$ into $\int \sqrt{1-x^2}\,\mathrm dx\;?$

10 votes
2 answers
416 views

Use Abel-Plana formula to compute $\int_0^\infty \frac{\arctan(t)}{e^{2\pi t}-1} dt $

9 votes
6 answers
723 views

A closed form for integral $\int_0^\infty \frac{\ln(1-e^{-\pi x})}{1+x^2} dx$

8 votes
2 answers
237 views

Derive a formula $\sum_{j=1}^{n-1}j^a \sim \zeta(-a)+\frac{n^{a+1}}{a+1}\sum_{s=0}^\infty \binom{a+1}{s}\frac{B_s}{n^s} $

8 votes
4 answers
234 views

If $|ax^2+bx+c|\le100$ for all $|x|\le 1$, What is the maxima for $|a|+|b|+|c|$

7 votes
0 answers
133 views

For any finite group $G$, is it always true that $ b_0\cdot b_1 \cdot\cdot\cdot b_{n}=b_0?$

6 votes
2 answers
302 views

Is there a general method to find the asymptotic order for this sequence?

5 votes
2 answers
167 views

Integral $\int_0^1 \frac{\ln(1-\sqrt{3}t+t^2)}{t} dt$

5 votes
3 answers
170 views

Integral $\int_0^1 \frac{(1+t^2)}{1-t^2+t^4}\ln(t)dt$

5 votes
2 answers
111 views

Is it possible to find a universal $\delta_U$, such that $|x-y|<\delta_U\Longrightarrow |f_n(x)-f_n(y)|<\epsilon, \forall n=\color{red}0,1,2\dots$

5 votes
2 answers
212 views

Why Wolfram gives inconsistent result for: $\int_0^\infty \frac{e^{x-x^2}-e^{-x-x^2}}{x}~dx$

4 votes
1 answer
92 views

If $z^{2023}=z+1$, prove $|z|\le 1$ if and only if the real part $\Re(z)\le -\frac{1}2$

4 votes
0 answers
89 views

A proof on the generating subset of a group

4 votes
6 answers
588 views

For this integral: $\int_0^{\frac{\pi}{4}} x \tan(x) dx$

3 votes
3 answers
88 views

$f(x)>0, f''(x)>0$. Prove: $\int_a^b f(x) dx > (b-a) f(\frac{a+b}{2})$

3 votes
1 answer
164 views

If $g$ is continuous and $A$ is a closed set and $A\subset \mathbb{R}$, then $g^{-1}(A)$ is closed

3 votes
0 answers
206 views

Abel-Plana formula for $\int_0^\infty \frac{f(x)}{e^{2\pi x}-1}dx$ and $\int_0^\infty \frac{f(x)}{\sinh(\pi x)}dx$

3 votes
1 answer
176 views

Compute $\int_0^\infty \frac{\sin(ax)}{b^2+x^2}~dx$, where $a>0, b>0$

3 votes
0 answers
84 views

Simplest form for $\int_0^\infty \frac{\sin(ax)}{b+x}~dx$

3 votes
0 answers
82 views

Is $G=\{ e\}$ both torsion group and a torsion-free group?

3 votes
0 answers
63 views

If $a_n=\sin(n^2)$, how to construct (if possible) a subsequence such that $\lim a_{n_k}\rightarrow 0$?

2 votes
2 answers
72 views

Is there a cyclic group $|\langle hk\rangle|= L$ or $|\langle hk\rangle|\ne L$?

2 votes
0 answers
59 views

Is it possible to prove $(G:H)\equiv (N:H) \pmod{p}$ without using Orbit Stabilizer Theorem or Sylow Theorem?

2 votes
1 answer
76 views

Proof Check: If $\phi_H=\{xH\mid hxH=xH, \forall h\in H\}$ and $N=\{g\in G| gHg^{-1}=H\}$ Prove: $|\phi_H|=(N:H)$

2 votes
1 answer
156 views

How to solve this non-linear differential equation.?

2 votes
1 answer
62 views

A series on Stieltjes constants, $\sum_{n=0}^\infty \frac{\gamma_n}{n!}$, where is the mistake?