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Proof that ${2x\over 2x-1}={A(x)\over A(x-1)}$ for every integer $x\ge2$, where $A(x)=\sum\limits_{n=0}^\infty\prod\limits_{k=0}^n\frac{x-k}{x+k}$

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How to study math to really understand it and have a healthy lifestyle with free time? [closed]

Real life applications of Topology

A multiplication algorithm found in a book by Paul Erdős: how does it work?

Am I just not smart enough? [closed]

If $a+b=1$ then $a^{4b^2}+b^{4a^2}\leq1$

Does associativity imply commutativity?

Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$

Is it an abuse of language to say "the integers," "the rational numbers," or "the real numbers," etc.?

How many resistors are needed?

Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?

Prove that $\sqrt{a^2+3b^2}+\sqrt{b^2+3c^2}+\sqrt{c^2+3a^2}\geq6$ if $(a+b+c)^2(a^2+b^2+c^2)=27$

If $a^3+a^2+a=9b^3+b^2+b$ and $a,b$ are integers then show $a-b$ is a perfect cube.

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

If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$

Prove that $\sum\limits_{cyc}\frac{a}{\sqrt{a+3b}}\geq\sqrt{a+b+c+d}$

What are examples of vectors that are not usually called vectors?

Evaluate $\frac{0!}{4!}+\frac{1!}{5!}+\frac{2!}{6!}+\frac{3!}{7!}+\frac{4!}{8!}+\cdots$

Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear.

Geometric inequality $\frac{R_a}{2a+b}+\frac{R_b}{2b+c}+\frac{R_c}{2c+a}\geq\frac{1}{\sqrt3}$

Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$

Inequality with Sum of Binomial Coefficients

Fastest way showing the limit exists without finding the limit?

Prove $ \sin x + \frac{ \sin3x }{3} + ... + \frac{ \sin((2n-1)x) }{2n-1} >0 $

Number as the sum of digits of some degree

Is $1 = [1] = [[1]] = [[[ 1 ]]], \ldots$?

Proof that ${2x\over 2x-1}={A(x)\over A(x-1)}$ for every integer $x\ge2$, where $A(x)=\sum\limits_{n=0}^\infty\prod\limits_{k=0}^n\frac{x-k}{x+k}$

$AB-BA$ is a nilpotent matrix if it commutes with $A$

Prove that between any two rational numbers there is a rational whose numerator and denominator are both perfect squares.

Determine all functions $f$ on $\mathbb R$ such that $f(x^2+yf(x))=f(x)f(x+y)$ for all $x,y$

$\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable