\( \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1} \)
\( \arctan (x) = \sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n+1}}{2n+1} \)
\( \arctan (1) = \frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{2n+1}\)
\( \pi = 6 \sum_{n=0}^{\infty} \frac{(2n)!}{2^{4n+1}(n!)^2 (2n+1)} \)
\( \frac{\pi}{2} = \prod_{n=1}^{\infty} \frac{4n^2}{4n^2-1} \)
Recursive relations with integration by parts shows that \( W_{n} = \int^{\pi}_{0} \sin^n (x)dx\) can be solved with the recursive series \(W_n = \frac{n-1}{n}W_{n-2}, W_0 = \pi, W_1 = 2\)
\(W_{2n} = \frac{2n-1}{2n}W_{2n-2}\) and that \(W_{2n+1} = \frac{2n}{2n+1}W_{2n-1}\).
By expanding \(W_{2n}\) out until its boundary conditions, one notices that \(W_{2n} = \pi \prod_{k=1}^{n} \frac{2k-1}{2k}\). Similarly, \(W_{2n+1} = 2 \prod_{k=1}^{n} \frac{2k}{2k+1}\).
Since sine has a maximum value of 1, \(W_{2n+1} \leq W_{2n} \leq W_{2n-1} \implies 1 \leq \frac{W_{2n}}{W_{2n+1}} \leq \frac{2n+1}{2n}\) (dividing by \(W_{2n+1}\) and applying reccurence relation) and by squeeze theorem, \( \lim_{n \to \infty} \frac{W_{2n}}{W_{2n+1}} = 1\)
\( \lim_{n \to \infty} \frac{W_{2n}}{W_{2n+1}} = 1 \implies \frac{pi}{2} \prod_{k=1}^{\infty} ( \frac{2k+1}{2k} \cdot \frac{2k-1}{2k} ) = 1 \), and by simple algebra the Wallis product is obtained.
\( \frac{2}{\pi} = \prod_{n=1}^{\infty} \cos(\frac{\pi}{2^{n+1}})\)
Sequence where the sum of the previous two values of the function determine the subsequent output
\(F_{n} = F_{n-1} + F_{n-2}\)
The converging limit between the ratio of two extremely large adjacent fibonacci numbers is the golden ratio, \(\varphi = \lim _{n \to \infty} \frac{F_{n}}{F_{n-1}}\). Why? \(\frac{F_{n+1}}{F_{n}} = \frac{F_n}{F_{n-1}}\) when \(n\) is infinitely large. This can be rearranged to \(\frac{F_n + F_{n-1}}{F_{n}} = \frac{F_n}{F_{n-1}}\), which fits the definition of the golden ratio
Consider \(f(x)=xe^x\), \(W(x)=f^{-1}(x)\),
Consider \(f(x)=xe^x\), \(W(x)=f^{-1}(x)\),
Function \(W\) as an inverse function for \(f(x)=xe^x\), however since \(f\) is not injective yet continuous, \(W\) is split into branches
\(W(z)e^{W(z)} = z\)
\(W_{0} : [-e^{-1}, \infty) \to [-1, \infty) \)
\(W_{0}(z)e^{W_{0}(z)} = z\)
\(W_{-1} : [-e^{-1},0) \to [-1, -\infty) \)
\(W_{-1}(z)e^{W_{-1}(z)} = z\)
\(f(x) = x - (1 + \frac{1}{x})^{x}\)
\(x_{n} = (1 + \frac{1}{x_{n-1}})^{x_{n-1}}\)
\( x_{n} = rx_{n-1} (1 - x_{n-1}) \)
\( \sum^{\infty}_{n=1} (\ln (e^n +1) - n) \)
\( \sum^{\infty}_{n=1} \sin (\frac{\pi}{2^n})\)
\( \sum^{\infty}_{n=1} \frac{n!}{n^n}\)
Notion applies to systems or processes that visit all parts in some space
\(m_n = \int^{\infty}_{-\infty} M_n(x) d\mu(x)\)
Mathematical model that maps each domain element in \(X\) to some codomain element in \(Y\). All \(x \in X\) have one mapping in \(Y\).
\(f: X \to Y\)
Set representing all elements that is a well-defined input for function \(f\)
\(\text{dom}(f) = \{ x \in X : f(x) \text{ is well defined } \}\)
Set representing the space that the image of \(f\) lies within
\(\text{codom}(f)\)
Set of all elements in a function's image mapped to elements in \(U\)
\(f(U) = \{ f(u) : u \in U \}\)
Also called the range, the subset of codomain representing all elements mapped to a domain element of function \(f\)
\( \text{im}(f) = f(\text{dom}(f)) \)
\( \text{im}(f) \subseteq \text{codom}(f) \)
Set of all elements in a function's domain mapped to elements in \(U\)
\(f^{-1}(U) = \{ u : f(u) \in U \}\)
\( \text{supp}(f) = \text{dom}(f) \setminus \text{ker}(f) \)
\((f \circ g)^{-1} = (g^{-1} \circ f^{-1}) \)
\( f \text{ is invertible} \iff f \text{ is injective} \)
\( f^{-1} : \forall x, f^{-1}(f(x))=x \)
\( f \text{ is invertible on } I \iff f \text{ is bijective on } I\)
\(f \text{ is linear} \iff \)
\( f \text{ is additive}\)
\( f \text{ is homogeneous of degree 1}\)
\( f \text{ is homogenous of degree }k \iff \forall \textbf{x} \forall c \gt 0 [ f(c \textbf{x}) =c^{k} f(\textbf{x}) ] \)
\(f \text{ is additive } \iff f(x+y) = f(x) + f(y)\)
\( \sum^{n}_{k=1} \lfloor \sqrt{k} \rfloor = -3S_1 ( \lfloor \sqrt{n} \rfloor ) + 2S_2 ( \lfloor \sqrt{n} \rfloor ) + 2\lfloor \sqrt{n} \rfloor + (n - \lfloor \sqrt{n} \rfloor^2)\sqrt{n}\)
\( \ln (x) = \lim_{n \to 0} \frac{x^n}-1{n} \)
\(R \text{ is strongly connected } \iff [ \forall x,y (xRy \lor yRx \lor x=y ) ]\)
\(\geq \text{ is a total order on }S \iff\)
\(\geq \text{ is a partial order on} S \iff\)
\(= \text{ is an equality relation on }S \iff\)
\(\Phi (x) = \int^{x}_{0} e^{-t^2}dt = \sum^{\infty}_{n=0} \frac{(-1)^n x^{2n+1}}{n!(2n+1)}\)
\( G = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^{2}} \)
\(G = \int_{0}^{1} \frac{\tan^{-1}(x)}{x}dx \)
\( f^{(-n)}(x) = \frac{1}{(n-1)!} \int_{0}^{x} (x-t)^{n-1}f(t)dt \)
By induction
\(A=\frac{\sum_{k=1}^{n} a_{k}}{n}\)
\(G=( \prod_{k=1}^{n} a_{k} )^{\frac{1}{n}}\)
\( S_{n} = \sum_{k=1}^{n}a+(k-1)d \)
\( S_{n} = \frac{n}{2} (2a+(n-1)d) \)
Consider a floor with planks of wood of length \(t\) and one drops sticks of length \(l\) onto the floor, \( l\leq t \implies \text{Pr}(\text{stick is across multiple planks}) = \frac{2}{\pi}\)
\( \displaystyle W_n = \int^{\frac{\pi}{2}}_{0} \sin^n (x) dx\)
\( W_n = \frac{n-1}{n} W_{n-2}\)
\( \text{li}(x) = \int^{x}_{0} \frac{1}{\ln (x)}dx \)
\(S(x)=\int^{x}_{0} \sin(t^2)dt\)
\(C(x)=\int^{x}_{0} \cos(t^2)dt\)
\(\text{Ti}_2 (x) =\sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)^2}x^{2n+1}\)
\(\text{Ti}_2 (x) = \int^{x}_{0} \frac{\tan^{-1}t}{t}dt\)
\( \chi_{\nu}(z) = \sum^{\infty}_{k=0} \frac{z^{2k+1}}{(2k+1)^{\nu}} \)
Entire functions have product of zeroes representation
Vector field that graphically describes solution of a first order ODE \(y' = f(x,y) \) by graphing tangents at any domain-codomain pair
\( \textbf{d}(x,y) = \begin{pmatrix} 1 \\ f(x,y) \end{pmatrix} \)
For differential equations that have readily known constant solutions (making \(y'=0\)), the differential equation can be remodelled as an algebraic one (like the transformation \(y'=k(a-y)(b-y),y \to f(x)=k(a-x)(b-x)\)) and since \(f(x)\) relates to \(y'\) a positive value at \(f(x)\) means that this solution tends away from the most recent root of \(f\). If \(f(x)\) is negative, this solution tends to the most recent root of \(f\)
Subset of the complex numbers of any number \(c\) that converges when processed through \(z_n\)
\(z_0 = 0\)
\(z_n = z_{n-1}^2 + c\)
\(f_c(z)=z^2+c\)
\(M= \{ c\in \mathbb{C} : \sup \{f_c(z)\} \lt \infty \}\)
Function that transforms a function into a new function by multiplying it by an integral kernel and integrating over some domain
\( \mathcal{T}\{f\}(y) = \int_{D} K(x,y)f(x)dx \)
Function used within an integral transform
\( K \text{ is a kernel of } \mathcal{T} \iff \mathcal{T}\{f\}(y) = \int_{D} K(x,y)f(x)dx \)
Function such that the distance between two range elements is bound by the difference between their domain mappings to the power of \(\alpha\) (up to a scaling constant); this is a generalization of the notion of a Lipschitz continuous function
\(f \text{ is } \alpha \text{-Hölder continuous } \iff \exists M \gt 0 : |f(x) - f(y)| \leq M|x-y|^{\alpha}\)
Family of subsets that contain the whole set of interest \(X\)
\(C = \{U_{\alpha} : \alpha \in A\} \text{ is a cover of }X \iff \bigcup_{\alpha \in A} U_{\alpha} \)
Point of a shape representing the average of each vertex
\(T = (\textbf{P}_1, \textbf{P}_2, \textbf{P}_3) \implies C = \frac{1}{3} \sum_{i=1}^{3} \textbf{P}_i\)
Ordered, enumerated collection of objects that allows for repetition, it is a generalization of a sequence to any ordered index set with total order rather than merely an interval of integers. it uses the same notation as sequences
Formally it is defined as a function \( (x_i) : I \to X\)
Sequence of sets where each set is a subset of the previous (other than the very first)
\( (U_{j})_{j \in J} \text{ is nested } \iff [ i \leq j \implies U_{i} \subseteq U_{j} ] \)
Ordered family of sub \(\sigma\)-algebrae that is non-decreasing; subsequent sets are supersets of all previous sets
\(\Sigma_{*} \text{ is a filtration on }(\Omega,\Sigma,\text{Pr}) \iff)\)