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  • Member for 1 year, 5 months
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22 votes
3 answers
1k views

Conjecture: $\sum\limits_{k=1}^nk^m=S_3(n)\times\frac{P_{m-3}(n)}{N_m}$ for odd $m>1 \ ;\ =S_2(n)\times\frac{P_{m-2}'(n)}{N_m}$ for even $m$.

17 votes
8 answers
775 views

Complex analysis book recommendations with more exercise than Ahlfors' book.

14 votes
3 answers
620 views

What is the least number of circles with radius $r$ is required to cover a circumference of a circle with radius $R>r$?

14 votes
7 answers
1k views

How to evaluate $\int_0^{\frac{\pi}{2}} \frac{\ln(\cos(x))}{1+\sin^2(x)} \, dx$

12 votes
2 answers
632 views

How to solve $\displaystyle\lim_{n\to\infty}\int_0^3\underbrace{\sin(\frac{\pi}{3}\sin(\frac{\pi}{3}...\sin(\frac{\pi}{3} x)...))}_\text{n sines}dx$?

11 votes
3 answers
449 views

How to evaluate $ \sum\limits_{k=0} ^{\infty} \frac{(-1)^k}{4k+3}$?

9 votes
2 answers
643 views

Can Stolz-Cesaro theorem be applied to this problem? If $\lim\limits_{x\to\infty}(f(x+1)-f(x))=l$, Prove that $\lim\limits_{x\to\infty}\frac{f(x)}x=l$

9 votes
3 answers
356 views

If $x^2-16\sqrt x =12$ what is the value of $x-2\sqrt x$?

8 votes
2 answers
239 views

What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

8 votes
3 answers
2k views

Does real analysis have new theorems, or is it just a collection of proofs of old calculus theorems?

7 votes
3 answers
3k views

What are the essential skills and abilities for pure mathematics? [closed]

7 votes
1 answer
191 views

How to solve $\int \frac{2020x^{2019}+2019x^{2018}+2018x^{2017}}{x^{4044}+2x^{4043}+3x^{4042}+2x^{4041}+x^{4040}+1}dx$

7 votes
2 answers
550 views

Is there is way to determine if the n-th roots of a polynomial is a polynomial?

6 votes
1 answer
115 views

Permutations of Triangle Centers: Investigating Relationships Between Circumcircle Intersections.

6 votes
1 answer
494 views

Why is Euclidean geometry neglected in modern study of mathematics? [closed]

6 votes
0 answers
239 views

Is there a purely algebraic proof that the sequence $(1+\frac{1}{k})^{k+\frac{1}{2}}$ is strictly decreasing? [duplicate]

6 votes
2 answers
137 views

How to represent $x^n$ as a sum of $P_k:= (x)(x-1)\dots(x-k+1)$?

6 votes
1 answer
212 views

Is it always true that if $x_n\to0$, $y_n\to0$ there exist $\epsilon_n\in\{-1,1\}$ such that both $\sum\epsilon_nx_n$and $\sum\epsilon_ny_n$ converge? [duplicate]

5 votes
2 answers
126 views

How to evaluate $\int\left(\frac{\sin(x)}{2\sin(x)- x(1+\cos(x))}\right)^2dx$?

5 votes
6 answers
616 views

What is $\sqrt{-1}$? circular reasoning defining $i$.

5 votes
1 answer
197 views

Conjecture: $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ only when $m $ is a prime number and $n$ is a power of $m$

5 votes
1 answer
184 views

If $\lim\limits_{x\to0}f(x)=0$ and $\lim\limits_{x \to 0}\frac{f(2x)-f(x)}{x}=0$ how to rigorously prove that $\lim\limits_{x \to 0}\frac{f(x)}{x}=0$? [duplicate]

5 votes
2 answers
117 views

How to evaluate$\lim\limits_{n\to\infty}n\left(n\left(n..\left(\int_0^1\left(\frac{\sqrt[n]{x}+1}{2}\right)^ndx-l_0\right)-l_1..\right)-l_{m}\right)$?

5 votes
1 answer
182 views

How to rigorously prove that $\sum\limits_{n=1}^ \infty( \frac{1}{4n-1} - \frac{1}{4n} )=\frac{\ln(64)- \pi}{8}$?

5 votes
1 answer
226 views

Closed form for $\Gamma(a-x)$ where $a \in (0,1]$.

5 votes
3 answers
844 views

What problems will arise if we define matrix multiplication this way?

4 votes
2 answers
232 views

Closed form for this generalisation of the gamma function. $f(x+1)=f(x)g(x+1) $

4 votes
1 answer
211 views

Is there a linear operator $T$ such that $T(x^n) =f(n)x^{n-1}$? [closed]

4 votes
0 answers
383 views

Are there Examples of a function with elementary antiderivative that we know its antiderivative is too big too be written down?

4 votes
4 answers
285 views

How to prove that $f(x)=\frac{ \cot(\frac{\pi}{x+1}) }{ \cot(\frac{\pi}{x}) }\cdot\frac{x}{x+1}$ is strictly deceasing for $x>2$?

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