Study Guide: Series, Sequences & Differential Equations in 111 Flashcards

Deck AP7391 — Series, Sequences & Differential Equations, 111 cards — is really two decks stitched together: cards 1–65 cover infinite series, cards 66–111 cover differential equations. Worth knowing before you start, because the study strategy for each half is completely different. Worth knowing too that the word "Sequences" in the title is a promise the deck does not keep: exactly one card mentions a sequence, and only as the sequence of partial sums. There is no card on monotone or bounded sequences, none on the squeeze theorem, none on recursive definitions.

The series half: a convergence-test arsenal

The strength of this deck is its test coverage. In cards 16–31 you get eleven of them: the nth-term (vanishing) test, direct comparison, limit comparison, ratio, root, integral, the alternating series (Leibniz) test with its error bound, Cauchy's condensation test, Dirichlet's test and Abel's test. Around them sit the named series every exam reuses — geometric, harmonic, alternating harmonic, telescoping, p-series, Grandi's, arithmetico-geometric — plus absolute versus conditional convergence, the Riemann rearrangement theorem, power series and radius of convergence, Taylor and Maclaurin series (with the five standard expansions), Laurent and Dirichlet series, the Riemann zeta function, the Basel problem, the Cauchy product, pointwise, uniform and compact convergence, the Weierstrass M-test, Fourier and asymptotic series, and the Cesàro and Abel summation methods.

The differential equations half: a taxonomy, not a toolbox

Cards 66–111 classify: ODE versus PDE, order, linear versus nonlinear, homogeneous versus inhomogeneous, initial value versus boundary value problems, existence and uniqueness (Peano, Picard–Lindelöf), then a gallery of named equations — Bernoulli, Euler–Lagrange, wave, heat, Laplace, harmonic oscillator, pendulum, KdV, Navier–Stokes — and the exotic variants (delay, stochastic, integro-differential, differential algebraic). What it never does is solve one. There is no separation of variables, no integrating factor, no characteristic equation, no undetermined coefficients, no Laplace transform.

Three tips that fit this deck specifically

Turn the eleven tests into decision cards. Definitions of the ratio test are not what an exam asks for. Rewrite each as a trigger: ratio test → factorials and nth powers; integral test → terms that are an integrable function of n; limit comparison → rational or algebraic terms; Leibniz → a visible (-1)ⁿ. Then drill in reverse: show yourself a series, name the test in five seconds.

Write the two endpoint cards yourself. Radius of convergence is here (card 33), interval of convergence is not — and every course problem asks for the interval, which means testing both endpoints by hand. Add one card for Σxⁿ/n at x = 1 and x = −1; that single pair is worth more marks than the Laurent and Dirichlet series cards combined.

Use the DE half for classification speed, not solving. Given any equation, say its order, whether it is linear, and whether it is homogeneous — under five seconds. That genuinely is an exam item, and it is exactly what these 46 cards train. Get your solving reps from problem sets instead.

Deck AP7391 is in the Flip It library at flip-it.app.

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