GRE Math Subject Test cheat sheet

Mathematics - GRE Math Subject Test

• Limit tests: ratio test gives convergence when lim |a_{n+1}/a_n| < 1, divergence when > 1, inconclusive at 1; root test uses lim |a_n|^{1/n} the same way; comparison and integral tests decide the inconclusive p-series cases.

• Derivative/integral facts: d/dx arcsin x = 1/sqrt(1-x^2), d/dx arctan x = 1/(1+x^2), d/dx a^x = a^x ln a; integral of sec x = ln|sec x + tan x| + C, integral of 1/(1+x^2) = arctan x + C.

• Taylor series worth memorizing: e^x = sum x^n/n!, sin x = sum (-1)^n x^{2n+1}/(2n+1)!, cos x = sum (-1)^n x^{2n}/(2n)!, 1/(1-x) = sum x^n for |x| < 1, ln(1+x) = sum (-1)^{n+1} x^n/n.

• Linear algebra: rank-nullity dim V = rank T + nullity T; a matrix is diagonalizable iff the sum of geometric multiplicities of eigenvalues equals n; det(AB) = det A det B; eigenvalues of A^k are lambda^k; trace equals the sum of eigenvalues.

• Group theory: Lagrange's theorem says |H| divides |G| for a subgroup H; a group of prime order is cyclic; the order of an element divides |G|; Z_m x Z_n is cyclic iff gcd(m,n) = 1; normality is needed to form G/H.

• Counting and probability: inclusion-exclusion |A∪B∪C| = sum |A| - sum |A∩B| + |A∩B∩C|; number of onto functions from an n-set to a k-set is sum (-1)^j C(k,j)(k-j)^n; expected value is linear regardless of dependence.

• Real analysis: a continuous function on a closed bounded interval attains its max and min (extreme value theorem); uniform convergence preserves continuity of the limit; every bounded sequence in R has a convergent subsequence (Bolzano-Weierstrass).

• Topology and complex analysis: continuous image of a compact set is compact; continuous image of a connected set is connected; a holomorphic function with zero derivative on a connected domain is constant; Cauchy-Riemann equations u_x = v_y, u_y = -v_x are necessary for differentiability.