Concept guide
Absolute convergence and the two certificates for alternating harmonic
Distinguish absolute convergence of a negative-ratio geometric series from conditional convergence of alternating harmonic.
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Concept derivation guides
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Concept guide
One data function under two parameter laws
Validate a total estimator and its parameterwise expectations.
Concept guide
Constructing a three-modulus CRT class
All integer solutions of x=2 mod 3, x=3 mod 5, x=2 mod 7 are 23+105t.
Concept guide
A spanning tree inside a square parent graph
Audit all vertices, edge membership, connectivity and acyclicity.
Concept guide
An exact L-squared tail on a finite law
Compute the second error moment and give a tail for any tolerance.
Concept guide
Unique decomposition and coverage in R3
R3=span{e1} direct-sum span{e2,e3}, with uniqueness and coverage separately verified.
Concept guide
Dual basis and covector change
For b1=(1,1),b2=(1,-1), phi1=(x+y)/2,phi2=(x-y)/2 form the dual basis.
Concept guide
Euclid and back-substitution for signed inputs
gcd(-84,30)=6, with (-84)*1+30*3=6.
Concept guide
An Euler circuit with an isolated vertex
Audit every edge exactly once and the nonisolated support condition.
Concept guide
A homeomorphism between finite chain spaces
Audit a relabeling homeomorphism using every open preimage.
Concept guide
A complete fibre-pushforward sampling distribution
Aggregate equal-valued outcomes of U under both theta laws.
Concept guide
A Chebyshev tail for fair Bernoulli averages
Derive the WLLN from a whole-sequence IID premise.
Concept guide
Excluding every two-color assignment on a triangle
Prove chi=3 using an upper witness and complete lower-color search.
Concept guide
All cliques and independent sets on three vertices
Classify all2³ subsets to obtain omega and alpha.
Concept guide
A bias table over the whole parameter set
Compute EthetaU-theta at every theta, distinct from observed error.
Concept guide
Same-law second and centered moment cross-checks
Compute variance at all theta values and verify centered definitions.
Concept guide
Fermat modulo 7 and a composite boundary
(-2)^6 is congruent to 1 modulo 7; for multiples of 7, only the congruence a^7≡a modulo 7 applies; passing a Fermat test does not prove primality.
Concept guide
All integer solutions with a negative coefficient
All integer solutions of -6x+9y=15 are x=5+3t,y=5+2t, t in Z.
Concept guide
Quotient by the diagonal line
R2/span{(1,1)} has basis [(0,1)] and dimension 1; its zero coset is the whole diagonal.
Concept guide
A complete six-element action on four points
The explicit 24-entry table is a valid group action, but it is neither free nor transitive.
Concept guide
Finite maxima and unattained interval suprema
Find sup/inf of A={-2,1/3,5/2} and B=(0,1), checking bounds and extremality.
Concept guide
A finite-chain basis and every generated open
Verify B={{a},{a,b},X} is a basis for the specified topology.
Concept guide
Deriving a Markov bound pointwise
Prove the finite-law bound and compare its exact tail.
Concept guide
Collision and ceiling load in a complete map
Use all fibres to prove a collision and the generalized lower bound.