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Analysis
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.
Lessons
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)
Other branches
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