MU-104 · Analysis I — Sequences & Series
Making limits rigorous: convergence, completeness, and the tests that tame infinite sums.
Lectures
| L01 | The Real Numbers: Fields and Order — |
| L02 | The Completeness Axiom: Suprema and Infima — |
| L03 | The Archimedean Property and Density of ℚ — |
| L04 | Sequences: Definition and Convergence — |
| L05 | Uniqueness of Limits and Boundedness — |
| L06 | The Algebra of Limits |
| L07 | Order Limit Theorems and the Squeeze |
| L08 | Monotone Sequences |
| L09 | The Monotone Convergence Theorem |
| L10 | Nested Intervals and e as a Limit |
| L11 | Subsequences and Limit Points — |
| L12 | The Bolzano–Weierstrass Theorem |
| L13 | Cauchy Sequences |
| L14 | Completeness of ℝ |
| L15 | Limsup and Liminf — |
| L16 | Series: Partial Sums and Convergence — |
| L17 | The Cauchy Criterion for Series |
| L18 | Series of Non-Negative Terms: Comparison — |
| L19 | The Ratio Test |
| L20 | The Root Test and Comparing the Tests |
| L21 | Alternating Series |
| L22 | Absolute versus Conditional Convergence |
| L23 | Rearrangement and the Riemann Series Theorem |
| L24 | Power Series and the Radius of Convergence |
| L25 | Sequences of Functions: Pointwise Convergence — |
| L26 | Synthesis: Completeness as the Engine of Analysis |
Theorems in this unit
The algebra of limits
Limits respect sums, products, and quotients.
The squeeze theorem
A sequence trapped between two convergents to L also converges to L.
The monotone convergence theorem
A bounded monotone sequence converges.
The Bolzano–Weierstrass theorem
Every bounded sequence has a convergent subsequence.
Completeness of the reals
A sequence of reals converges if and only if it is Cauchy.
The ratio test
A series converges absolutely when the limiting ratio of terms is below one.
The alternating series test
An alternating series with terms decreasing to zero converges.
The Riemann rearrangement theorem
A conditionally convergent real series can be rearranged to converge to any prescribed value, or to diverge to plus or minus infinity.
The Cauchy-Hadamard theorem
A power series converges absolutely inside the disc of radius R = 1/limsup |c_n|^(1/n) and has unbounded terms outside it; the circle is undecided.