Mathematics / core
Sequences, series, and limits of functions
Track how infinite processes interact with differentiation, integration, approximation, and functional limits.
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Module 01 core
Sequences, series, and limits of functions
Track how infinite processes interact with differentiation, integration, approximation, and functional limits.
- 01
Limsup, liminf, and subsequences
Extract persistent upper and lower behavior from an oscillating sequence without pretending it has a single limit.
29 min - 02
Comparison tests and positive series
Turn monotone partial sums into reliable comparison, condensation, and remainder estimates.
29 min - 03
Absolute and conditional convergence
Separate cancellation-dependent convergence from summability that survives changes of sign and ordering.
29 min - 04
Rearrangements and infinite products
Identify which infinite operations preserve a value and translate nonzero product convergence into logarithmic sums.
29 min - 05
Power series and their radius of convergence
Use coefficient growth to locate an interval of analytic control while checking its boundary independently.
29 min - 06
Pointwise and uniform convergence
Read the quantifier order that distinguishes local agreement from a simultaneous error guarantee.
29 min - 07
Exchanging limits and integrals
Match the mode of convergence to the measure, interval, and tail control required by integration.
29 min - 08
Differentiating series and function limits
Control derivatives and one anchor value before identifying the derivative of a limiting function.
29 min - 09
Equicontinuity and the Arzelà–Ascoli theorem
Find uniformly convergent subsequences by controlling both the heights and the shared spatial variation of functions.
29 min - 10
Polynomial density and Stone–Weierstrass
Understand uniform approximation as density of a function algebra, with exact hypotheses and no hidden Taylor assumption.
29 min