GRE Math Subject Test flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Ratio test conclusion

For a series with positive terms, compute L = lim |a_{n+1}/a_n|. If L < 1 the series converges absolutely, if L > 1 it diverges, and if L = 1 the test gives no information.

p-series convergence

The series sum 1/n^p converges exactly when p > 1 and diverges when p is at most 1. This is the standard benchmark for comparison and integral tests.

Alternating series test

If b_n decreases to zero, then sum (-1)^n b_n converges. The error after N terms is bounded by the first omitted term b_{N+1}.

Derivative of arctan

The derivative of arctan x is 1/(1 + x^2) for all real x. Correspondingly, the integral of 1/(1 + x^2) equals arctan x plus a constant.

Derivative of arcsin

The derivative of arcsin x equals 1 divided by the square root of (1 - x^2), defined for x strictly between -1 and 1.

Integral of sec x

The antiderivative of sec x is ln|sec x + tan x| + C. Differentiating this expression returns sec x by the chain rule and a tangent-secant identity.

Taylor series for e^x

e^x equals the sum over n of x^n divided by n factorial, valid for every real x. Truncating after N terms leaves an error bounded by the next term for positive x.

Geometric series sum

The sum from n equals zero to infinity of r^n equals 1/(1 - r) when |r| < 1, and diverges otherwise. This underlies the series for 1/(1 - x).

Rank-nullity theorem

For a linear map T from V to W with V finite dimensional, dim V equals the rank of T plus the nullity of T, where nullity is the dimension of the kernel.

Diagonalizability criterion

An n by n matrix over a field is diagonalizable exactly when the sum of the geometric multiplicities of its distinct eigenvalues equals n, equivalently when it has n independent eigenvectors.

Determinant multiplicativity

For square matrices of the same size, det(AB) equals det(A) times det(B). Consequently det(A inverse) is the reciprocal of det(A) when A is invertible.

Trace and eigenvalues

The trace of a square matrix equals the sum of its eigenvalues counted with algebraic multiplicity, and the determinant equals their product, both over an algebraic closure.

Lagrange's theorem

If H is a subgroup of a finite group G, then the order of H divides the order of G. Hence any group of prime order is cyclic and has no proper nontrivial subgroups.

Order of an element

The order of an element g is the smallest positive k with g^k equal to the identity. By Lagrange's theorem this order divides the order of the group.

Cyclic product of cyclic groups

The direct product Z_m x Z_n is cyclic, hence isomorphic to Z_mn, exactly when m and n are relatively prime. Otherwise its exponent is the least common multiple.

Quotient group condition

A subgroup H of G yields a well-defined quotient group G/H exactly when H is normal, meaning gHg inverse equals H for every g in G.

Inclusion-exclusion

For three finite sets, the size of the union equals the sum of individual sizes minus the pairwise intersection sizes plus the triple intersection size.

Onto function count

The number of surjections from an n-element set to a k-element set is the alternating sum over j of (-1)^j C(k,j)(k-j)^n, derived from inclusion-exclusion on missing images.

Linearity of expectation

The expected value of a sum of random variables equals the sum of their expected values, with no independence assumption required. Variance additivity, by contrast, requires uncorrelated variables.

Extreme value theorem

A continuous real-valued function on a nonempty closed bounded interval attains both a maximum and a minimum value somewhere on that interval.

Intermediate value theorem

If f is continuous on a closed interval and y lies between f at the endpoints, then some point in the interval has value y.

Uniform convergence preserves continuity

If a sequence of continuous functions converges uniformly to a limit function on a set, that limit is continuous. Pointwise convergence alone does not guarantee continuity of the limit.

Bolzano-Weierstrass theorem

Every bounded sequence of real numbers contains a convergent subsequence. Equivalently, every bounded infinite subset of the reals has a limit point.

Compactness preservation

The continuous image of a compact topological space is compact. In the reals this yields the extreme value theorem for closed bounded intervals.

Connectedness preservation

The continuous image of a connected space is connected. In the reals, connected subsets are exactly the intervals, which yields the intermediate value theorem.

Cauchy-Riemann equations

For f equal to u plus iv to be complex differentiable, u_x equals v_y and u_y equals -v_x must hold. Together with continuous partials, these conditions are also sufficient.

Zero derivative implies constant

If a holomorphic function has derivative zero at every point of a connected open set, then the function is constant on that set.

Radius of convergence

For a power series in x minus a, the radius R satisfies 1/R equals limsup of the nth root of the coefficient's absolute value; convergence is absolute inside and divergent outside.

Cauchy-Schwarz inequality

For vectors in an inner product space, the absolute value of the inner product is at most the product of the norms. Equality holds exactly when the vectors are linearly dependent.

Euler's formula for planar graphs

For a connected planar graph drawn without crossings, the number of vertices minus edges plus faces equals 2, counting the unbounded outer face.

Mean value theorem

If f is continuous on a closed interval and differentiable inside it, then some interior point has derivative equal to the average slope across the whole interval.

Fundamental theorem of calculus

If f is continuous, the function defined by integrating f from a to x has derivative f at x. Consequently the definite integral equals F(b) minus F(a) for any antiderivative F.