Sequences Portal

1Learning Path

This portal introduces concepts in the order required within the sequence series. Prerequisites outside this series are specified in each lecture's prerequisites. The eight lecture cards follow the recommended order in _order.json; the exercise card is placed separately at the end.

21. Overview and Foundations

First establish indices, explicit formulas, recurrences, and unique determination by initial conditions. Then define arithmetic and geometric sequences through relations between adjacent terms and learn their general terms and finite sums.

32. Operators and Classification

Use the shift operator and forward difference to express recurrences as operator equations. Then classify them by order and linearity, and organize the criteria for selecting differences, equilibria, characteristic polynomials, or state vectors.

43. Specific Solution Methods

The linked lecture titled “First-Order Recurrences” treats linear recurrences of order one: separate $p=1$p=1 from $p\ne 1$p\ne1 in $a_{n+1}=p{a}_n+q$a_{n+1}=pa_n+q and prove homogenization by an equilibrium point. For difference equations, develop the formulation $\mathrm{\Delta }=E-I$\Delta=E-I and its connection with linear operator equations.

54. Limits and Series

Define convergence of sequences by the $\epsilon$\varepsilon-$N$N criterion and study divergence and tests involving geometric sequences, rational expressions, and squeezing. Apply that definition to sequences of partial sums to analyze convergence of infinite series, its necessary condition, and geometric series.

65. Exercises

These exercises consolidate classification and selection of solution methods. They may be attempted after completing the operators-and-classification stage; the lectures on limits and series are not prerequisites.