analysis i
The course with the biggest jump up from school maths.
It started by asking questions that felt obvious: what is a real number? What is addition? I remember attempting the first problem sheet and proving from the axioms things like , or , thinking what have I got myself into. But that's part of why I love maths: you prove things once, properly, and then cache them in your toolkit to prove bigger results. In this course, that meant convergence of sequences and the series & power series built out of them.
syllabus
Real numbers: arithmetic, ordering, suprema, infima; the real numbers as a complete ordered field. Definition of a countable set. The countability of the rational numbers. The reals are uncountable. The complex number system. The triangle inequality.
Sequences of real or complex numbers. Definition of a limit of a sequence of numbers. Limits and inequalities. The algebra of limits. Order notation: , .
Subsequences; a proof that every subsequence of a convergent sequence converges to the same limit; bounded monotone sequences converge. Bolzano–Weierstrass Theorem. Cauchy's convergence criterion.
Series of real or complex numbers. Convergence of series. Simple examples to include geometric progressions and some power series. Absolute convergence, Comparison Test, Ratio Test, Integral Test. Alternating Series Test.
Power series, radius of convergence. Examples to include definition of and relationships between exponential, trigonometric functions and hyperbolic functions.