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Real Analysis

Limits made rigorous: sequences, series, convergence, improper integrals, complex numbers as functions.

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Core Sequences and their limits What it means for aₙ to approach L, and the limits that define e and π. limit of (1 + 1/n)^n as n -> oo Core Convergence of series Divergence, ratio, comparison and integral tests, and the harmonic series. does 1/n^2 for n = 1 to oo converge Core Improper integrals Integrals to infinity or across a singularity, defined as limits. integrate 1/x^2 dx from 1 to oo Advanced Taylor approximation and error How well a polynomial stands in for a function, and how far you can trust it. taylor series of sin(x) Advanced Functions of a complex variable Euler's formula, and why complex analysis is where the Riemann Hypothesis lives. e^(i*pi) Core Continuity and differentiability, rigorously ε, δ continuity, the intermediate and extreme value theorems, the mean value theorem. limit of sin(x)/x as x -> 0 Advanced Sequences of functions and uniform convergence Pointwise versus uniform, and when limits commute with integrals and derivatives. taylor series of e^x

Analysis is calculus with the reasons attached: what a limit really is, when a series converges and why, what makes a function continuous or differentiable, and how the same ideas extend to the complex plane. It is the language the open problems are written in.

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Sequences and series

Real analysisSequenceLimit of a sequenceCauchy sequenceSeries (mathematics)Convergent seriesAbsolute convergenceRatio testComparison testAlternating series testPower seriesRadius of convergenceHarmonic series (mathematics)

Functions

Limit of a functionContinuous functionUniform continuityIntermediate value theoremExtreme value theoremMean value theoremDifferentiable functionTaylor's theoremRolle's theoremL'Hôpital's ruleMonotonic function

The real line

Real numberCompleteness of the real numbersSupremumInfimumBolzano–Weierstrass theoremHeine–Borel theoremDense setRiemann integralImproper integralFundamental theorem of calculusWeierstrass functionStone–Weierstrass theorem

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What is the ε, δ definition actually saying?

That you can make the output as close to the limit as anyone demands (within ε) by keeping the input close enough (within δ). It replaces "approaches" with a challenge-and-response that can be checked.

Why does the harmonic series diverge when its terms go to zero?

Because the terms shrink too slowly: group them as 1/3 + 1/4 > 1/2, 1/5 + … + 1/8 > 1/2, and so on: infinitely many halves.

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