Cambridge Part IA Analysis I: Syllabus, Topics and Practice

Analysis I is the first rigorous analysis course in Part IA, the first year of the Cambridge Mathematical Tripos. It runs for 24 lectures in Lent Term and is examined in Paper 1, alongside Vectors and Matrices. It takes the calculus you already know and proves it properly, from convergent sequences to the Riemann integral. This guide follows the official 2026-27 Faculty of Mathematics schedules and links every topic to free practice.

What Cambridge Analysis I covers

The schedules list five blocks, with approximate lecture counts:

1. Limits and convergence [6]. Sequences and series in \mathbb{R} and \mathbb{C}, absolute convergence, the Bolzano-Weierstrass theorem and the General Principle of Convergence, and the comparison, ratio and alternating series tests.
2. Continuity [3]. Continuity of real and complex functions, the Intermediate Value Theorem, and the fact that a continuous function on a closed bounded interval is bounded and attains its bounds.
3. Differentiability [5]. Derivatives of sums and products, the chain rule, the derivative of an inverse function, Rolle's theorem, the Mean Value Theorem, a one-dimensional inverse function theorem, Taylor's theorem with Lagrange remainder, and complex differentiation.
4. Power series [4]. Complex power series and radius of convergence, and the exponential, trigonometric and hyperbolic functions and how they relate.
5. Integration [6]. The Riemann integral, a non-integrable function, integrability of monotonic and piecewise-continuous functions, the Fundamental Theorem of Calculus, integration by parts, the integral form of Taylor's remainder, and improper integrals.

The schedules list no formal prerequisites. In North American terms the results are those of Calculus 1 and the series part of Calculus 2, but each one is proved rather than only applied.

Analysis I topic map

Example problems at the Analysis I level

1. The ratio test. For \sum_{n \ge 1} \frac{n}{2^n},

so the series converges absolutely.

2. Radius of convergence. For \sum_{n \ge 1} \frac{z^n}{n}, the ratio \left|\frac{a_{n+1}}{a_n}\right| = \frac{n}{n+1}|z| \to |z|, so the radius of convergence is R = 1.

3. Using the Mean Value Theorem. If f'(x) = 0 for all x in an interval, then for any a < b in it there is a c with

so f is constant. Results like this are why the MVT appears in so many Analysis I proofs.

How to do well in Analysis I

• Write out the definition first. Almost every proof begins by stating what convergence, continuity or differentiability means in symbols.
• Know a counterexample for each theorem. Why does the IVT need continuity? Why does the ratio test say nothing when the limit is 1? The harmonic series and the absolute value function are good places to start.
• Keep computations automatic. Fluent convergence tests and Taylor series leave your attention free for the logic.
• Review the Mean Value Theorem guide before the exam. It drives a large share of Paper 1 questions on differentiation.

Tip: Differential Equations in Michaelmas uses Taylor series and l'Hôpital's rule informally, and Analysis I proves them. See the Cambridge Differential Equations guide.