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Definitions
$x\text{ is even} := \exists y[y+y=x]$ (Denote as $E(x)$)
$x\text{ is odd} := \exists y[y+y+1=x]$ (Denote as $O(x)$)
Axioms
$\forall x[S(x) \ne 0]$
$\forall x \forall y[S(x)=S(y) \implies x=y]$
$\forall x[x+0=x]$
$\forall x \forall y[x+S(y)=S(x+y)]$
$[\varphi(0) \land [\forall x...
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$${ \sum_{r=1}^{\infty}\frac{(2r)!!}{(2r+3)!!}}$$
What approach do we need to solve such type of summations?
My attempt:
\begin{align}
\sum_{r=1}^{\infty}\frac{(2r)!!}{(2r+3)!!}
&= \sum_{r=1}^{\infty}\frac{(2r)!!\cdot (2r+2)!!}
{(2r+3)!!(2r+2)!!} \\
&= \sum_{r=1}^{\infty}\frac{2^r\cdot (r)!\cd...
In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems) and connected problems like representation and duality. Well known results like the representation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic (Czelakowski 2003).
Works...
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.
Traditionally, algebraic statistics has been associated with the design of experiments and multivariate analysis (especially time series). In recent years, the term "algebraic statistics" has been sometimes restricted, sometimes being used to label the use of algebraic geometry and commutative algebra in statistics.
== The tradition of algebraic statistics ==
In the past, statisticians have used algebra to advance research in statistics...
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0. In other words, whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. The theorem was first stated by Henri Poincaré...
15
$$\frac{\pi^2}{24}-\frac{2\pi}3+\frac1{36\sqrt{2}}\left[5\pi^2+12\left(4+\ln\left(\frac{1+\sqrt2}2\right)\right)\left(\pi-2\ln\left(1+\sqrt2\right)\right)-48\operatorname{Li}_2\left(\sqrt2-1\right)\right]$$
