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GohP.iHan
  • Member for 9 years, 11 months
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26 votes
2 answers
8k views

Closed form for $ S(m) = \sum_{n=1}^\infty \frac{2^n \cdot n^m}{\binom{2n}n} $ for integer $m$?

17 votes
3 answers
653 views

Bounds of $1^n + 2^{n-1} + 3^{n-2} + \cdots + n^1$

13 votes
1 answer
8k views

$a,b,c,d\ne 0$ are roots (of $x$) to the equation $ x^4 + ax^3 + bx^2 + cx + d = 0 $

8 votes
1 answer
2k views

Prove (or disprove): $a\times b=c\times d$, the solutions for $x$ in the equation $\frac {a^x+b^x}{c^x+d^x} = \frac{a+b}{c+d}$ is only $\pm 1$.

6 votes
1 answer
245 views

True? $\sum_{n=1}^\infty {n+m-1 \choose m}^{-1} = 1 + \frac {1}{m-1} $

5 votes
4 answers
313 views

A caboodle of Pell's equation in one? $x^2+y^2-5xy+5=0$

4 votes
1 answer
111 views

Is there a formula for the closed form for $ \displaystyle \sum_{r=1}^\infty \frac{\sum_{k=1}^r k^n}{r!}$ for any positive integer $n$?

4 votes
1 answer
315 views

Other variation of Nicomachus's Theorem?

4 votes
0 answers
822 views

Minimize $f(m)=\sum_{n=1}^\infty n^m / m^n $

4 votes
3 answers
956 views

Prove that $\zeta(-1)=\zeta(-13)$.

3 votes
1 answer
201 views

Find all integer solutions for $ \frac{B_{2m}}m =\frac{B_{2n}}n$.

3 votes
2 answers
84 views

Generalized sum of distinct integers $ \sum_{\stackrel{a_1,a_2,\ldots, a_n>0}{\text{all distinct}}} 2^{-\sum a_j} $

2 votes
0 answers
58 views

Jacobian transformation on $ \int\cdots \int \frac{x_1 + x_2 + \cdots + x_n }{ \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2 } } dx_1 dx_2 \cdots dx_n $

2 votes
10 answers
249 views

Proving a convoluted proof to an inequality: $x,y>0$ such that $x^2+y^2=1$. Prove that, $x^3+y^3 \geqslant \sqrt2 xy $

2 votes
3 answers
379 views

Prove that $ \int_0^1 x^2 \psi(x) \, dx = \ln\left(\frac{A^2}{\sqrt{2\pi}} \right) $

2 votes
1 answer
632 views

Prove this inequality $ \sqrt{5} > \frac {13 + 4\pi}{24 - 4\pi} $ [closed]

2 votes
2 answers
98 views

Can $ 2^{3^{4^{.^{.^{.^{n-1}}}}}}\equiv 1 \bmod {n} $ for some $n>7$.

1 vote
0 answers
35 views

Least degree polynomial and Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=2,3,\ldots,n+1$.

1 vote
1 answer
131 views

Simplified form for a Newton's sum? $a_1^k + a_2 ^k + \ldots + a_n^k = k $ for $k=1,2,\ldots,n$.

1 vote
1 answer
106 views

Hidden random walk in shallow sums? Prove that $\sum_{k=0}^n k \cdot \binom{2n-k}n 2^k = (2n+1) \cdot \binom{2n}n - 4^n $

1 vote
1 answer
144 views

Derangement probability distribution question: Why $\text{Var}[x] \approx E[X]$ is fulfilled?

1 vote
1 answer
132 views

Prove (or disprove) that $ \sum_{n=1}^\infty \frac{4(-1)^n}{1-4n^2} x^n = \frac{2(x+1) \tan^{-1}(\sqrt x)}{\sqrt x} - 2 $ for $ 0<x\leq1$

1 vote
1 answer
11k views

Elementary solution to $ \int \frac{1}{x^5+1} \, dx $ [duplicate]

0 votes
1 answer
84 views

Derangement question but you stop once you've found a matching pair

0 votes
1 answer
32 views

The high occurence of a number equals to the last few digits of its own square

0 votes
2 answers
147 views

prove $ \displaystyle \lim_{x \to 0} (1+x)^{\frac {1}{x} } = e $ by epsilon delta?