KCL 4CCM115A Sequences and Series: Syllabus, Topics and Practice

4CCM115A Sequences and Series is a Year 1 autumn module in the Department of Mathematics at King's College London and one of the required first-year modules in the Mathematics BSc. It is where many students meet proof-based mathematics for the first time. This guide sums up what King's says the module covers and links each topic to free practice, so the computations behind the proofs become second nature.

What KCL Sequences and Series covers

King's describes the aim as introducing the ideas and methods of university-level pure mathematics: showing why proofs are needed, encouraging logical arguments and conveying the power of abstract methods. This is done through one connected story: real numbers, sequences, limits and series.

The official syllabus has three parts:

1. Sets and functions: defining sets, subsets, intersections and unions; injections, surjections and bijections; compositions and inverses of functions.
2. Logic and proof: logical operations, implication, equivalence, quantifiers, converse and contrapositive; proof by induction and by contradiction.
3. Sequences and series: applying these ideas to the real numbers to make rigorous arguments about sequences and series, and to develop convergence and limits.

The learning outcomes say you should be able to compute limits of sequences involving elementary functions and prove simple statements involving convergence. In calculus terms, the third part lines up with the sequences and series unit of Calculus 2, studied with full rigour.

4CCM115A topic map

Proofs are best learned from the module itself, but the calculations underneath them can be practised here.

The module page does not list specific convergence tests by name, so treat the test rows as useful background rather than a promise of exam content.

How it fits with Calculus I

The two autumn modules support each other. Calculus I gives you the techniques, and this module explains why they work. In the spring, Calculus II moves into higher dimensions.

Example problems at the 4CCM115A level

1. A limit with a proof. To show \frac{1}{n} \to 0, take any \varepsilon > 0 and choose N > \frac{1}{\varepsilon}. Then for every n > N you have \left|\frac{1}{n} - 0\right| < \varepsilon.

2. A geometric series. For |r| < 1, the partial sums are S_n = \frac{1 - r^n}{1 - r}, so

3. A divergent series. The harmonic series \sum \frac{1}{n} diverges even though its terms tend to 0. Grouping terms shows S_{2^m} \ge 1 + \frac{m}{2}.

4. Proof by induction. Show that 1 + 2 + \dots + n = \frac{n(n+1)}{2} for every n \ge 1.

How to do well in Sequences and Series

• Learn the definitions word for word. In this module the definition of convergence is the starting point of almost every proof.
• Do the weekly problems every week. King's discusses them in tutorials, and they are the closest guide to the class tests.
• Keep the computations easy. If finding a limit of a sequence still takes effort, practise with the sequences lesson so the proof is the only hard part.
• Write proofs in full sentences. Markers look for clear logic, not just the right number. Our study tips cover steady weekly habits.

Tip: when a series problem looks new, try comparing it with a geometric series or the harmonic series first. Many convergence arguments come down to one of those two.