Formulas & Methods

Limits as derivatives ($n(a^{1/n} - 1) \to \ln a$), as Riemann sums ($\tfrac1n\sum f(\tfrac kn) \to \int_0^1f$), as exponentials ($(1 + \tfrac an)^n \to e^a$); Stolz–Cesàro; Taylor with Lagrange remainder; the three series ($\sum x^n$, $\sum\tfrac{x^n}{n}$, $\sum\tfrac{x^n}{n!}$) and their derivatives; $\zeta(2) = \tfrac{\pi^2}{6}$, $\eta(2) = \tfrac{\pi^2}{12}$; telescoping with lag; Cauchy condensation; Abel summation $\sum a_nb_n = A_Nb_N - \sum A_n(b_{n+1} - b_n)$.

Feynman: $I'(a) = \int\partial_af$; Dirichlet $\int_0^\infty\tfrac{\sin x}{x} = \tfrac\pi2$ and $\int_0^\infty\tfrac{\sin^2x}{x^2} = \tfrac\pi2$; Gaussian $\int e^{-x^2} = \sqrt\pi$ and $\int_0^\infty e^{-x^2}\cos 2bx = \tfrac{\sqrt\pi}{2}e^{-b^2}$; King $\int_a^bf = \int_a^bf(a+b-x)$; Queen/Jack folding; Glasser $\int f(x - \tfrac1x) = \int f$; the $1/x$ reflection; Frullani $\int_0^\infty\tfrac{f(ax) - f(bx)}{x} = (f(0) - f(\infty))\ln\tfrac ba$; Gamma $\Gamma(n+1) = n!$, $\Gamma(\tfrac12) = \sqrt\pi$, $\Gamma(s)\Gamma(1-s) = \tfrac{\pi}{\sin\pi s}$; Beta $B(p,q) = \tfrac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}$; $\int_0^\infty\tfrac{dx}{1+x^n} = \tfrac{\pi/n}{\sin(\pi/n)}$; log-sine $\int_0^{\pi/2}\ln\sin x = -\tfrac\pi2\ln2$; $\int_0^1\tfrac{\ln(1+x)}{1+x^2} = \tfrac\pi8\ln2$; $\int_0^\infty\tfrac{\ln(1+x^2)}{1+x^2} = \pi\ln2$; Fresnel $\int_0^\infty\sin x^2 = \tfrac12\sqrt{\tfrac\pi2}$; Wallis $\prod\tfrac{4n^2}{4n^2-1} = \tfrac\pi2$; Weierstrass $t = \tan\tfrac x2$; series expansion and swap.

AM–GM, Cauchy–Schwarz (sums and integrals), Jensen, rearrangement, power means; Hermite–Hadamard $f(\tfrac{a+b}{2}) \le \tfrac{1}{b-a}\int_a^bf \le \tfrac{f(a) + f(b)}{2}$ for convex $f$; tangent-line trick; MVT-derived bounds ($e^x \ge 1 + x$, $\ln x \le x - 1$, $\sin x \le x$).

Matrix determinant lemma $\det(I + \mathbf u\mathbf v^{\mathsf T}) = 1 + \mathbf v^{\mathsf T}\mathbf u$ and $\det(aI + bJ) = a^{n-1}(a + nb)$; Vandermonde $\prod_{i<j}(x_j - x_i)$; circulant eigenvalues by DFT; tridiagonal Toeplitz eigenvalues $a + 2b\cos\tfrac{k\pi}{n+1}$; $\det p(A) = \prod p(\lambda_i)$, $\mathrm{tr}A^k = \sum\lambda_i^k$; Cayley–Hamilton; rank–nullity, $\mathrm{rank}(AB) \le \min$; polynomial-entry matrices have rank $\le \deg + 1$; $|GL_n(\mathbb F_q)| = \prod(q^n - q^k)$; spectral theorem and Rayleigh quotient; $\mathrm{tr}A^k = 0$ for all $k \le n$ implies nilpotent.

Vieta and Newton; roots of unity filter; auxiliary polynomial with known roots for values at integers; Chebyshev; Eisenstein; $a - b \mid P(a) - P(b)$. Orders and CRT; LTE; Legendre/Kummer; QR supplements and reciprocity; Wilson; multiplicative functions and $\sum_{d\mid n}\varphi(d) = n$; bounding between squares. Lagrange's theorem; cyclic-group counts ($\varphi(d)$ elements of order $d$, one subgroup per divisor); orders in $S_n$ as lcm of cycle lengths; a polynomial over a field has at most $\deg$ roots; finite closed subsets are subgroups.

Bijections; PIE; derangements; Catalan and the reflection principle; generating functions and $x\tfrac{d}{dx}$; Erdős–Szekeres; Mantel/Turán, Hall, Euler; extremal and pigeonhole; games (P/N, Nim, strategy stealing). Linearity with indicators; tail-integral $E[X] = \int P(X > t)dt$; order statistics $E[\max] = \tfrac{n}{n+1}$; coupon collector $nH_n$; gambler's ruin $\tfrac kN$, duration $k(N-k)$; random walk returns $\binom{2n}{n}/4^n$; first-step analysis.

Pick's theorem; $\gcd + 1$ lattice points on a segment; Minkowski; Gauss circle count $\pi r^2 + O(r)$; complex numbers (rotation, collinearity via real ratios, equilateral criterion $a + \omega b + \omega^2c = 0$); point-to-line/plane distances; convexity (Helly, Carathéodory); continuous-motion IVT arguments.