M2: Analysis I - Sequences and Series (2025-26)
Main content blocks
- Lecturer: Profile: Richard Earl
\(\bullet\) prove results within an axiomatic framework;
\(\bullet\) define and prove basic results about countable and uncountable sets, including key examples;
\(\bullet\) define what it means for a sequence or series to converge;
\(\bullet\) prove results using the completeness axiom for \(\mathbb{R}\) and using Cauchy’s criterion for the - convergence of real and complex sequences and series, and explain how completeness and Cauchy’s criterion are related;
\(\bullet\) analyse the convergence (or otherwise) of a variety of well known sequences and series, and use this to conjecture the behaviour of unfamiliar sequences and series;
\(\bullet\) apply standard techniques to determine whether a sequence converges, and whether a series converges;
\(\bullet\) define the elementary functions using power series, and use these definitions to deduce basic properties of these functions.
Sequences of real or complex numbers. Definition of a limit of a sequence of numbers. Limits and inequalities. The algebra of limits. Order notation: \(O\), \(o\).
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.
Section outline
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This sheet studies the consequences of the field and order axioms from lectures 1-2.
There are optional exercises introducing other fields.
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This sheet includes exercises on completeness, suprema and infima, and countability from lectures 3-4.
In the optional exercises an alternative for the completeness axiom is discussed.
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We introduce sequences and the notions of convergence and limit. Optional questions discuss convergence more generally in metric spaces.
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Exercises on the algebra of limits and monotone sequences. We further introduce e and show it is irrational. An optional question shows that \( \pi \) is irrational.
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Exercises on subsequences and the Cauchy convergence criterion. The optional exercises introduce the notion of double sequences.
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Exercises on infinite sums - their analysis requires the different convergence tests we have met. Mercator's series is shown to be conditionally convergent. The optional exercises introduce infinite products.
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The exercises are on power series and finding radii of convergence. Applications include solving differential equations, generating functions and a little analytic number theory.