Unit · year 1
MU-104 · Analysis I — Sequences & Series
Threads change · space7 theorems
Making limits rigorous: convergence, completeness, and the tests that tame infinite sums.
Theorems in this unit
T-021
The algebra of limits
Limits respect sums, products, and quotients.
T-022
The squeeze theorem
A sequence trapped between two convergents to L also converges to L.
T-023
The monotone convergence theorem
A bounded monotone sequence converges.
T-024
The Bolzano–Weierstrass theorem
Every bounded sequence has a convergent subsequence.
T-025
Completeness of the reals
A sequence of reals converges if and only if it is Cauchy.
T-026
The ratio test
A series converges absolutely when the limiting ratio of terms is below one.
T-027
The alternating series test
An alternating series with terms decreasing to zero converges.