# Cambridge Part IA Analysis I: Syllabus, Topics and Practice

> A study guide for Cambridge Part IA Analysis I: convergence tests, continuity, the Mean Value Theorem, power series and the Riemann integral, with practice.

Source: https://practicecalculus.com/courses/university-of-cambridge-analysis-i
Updated: 2026-11-08

## At a glance

- **Course code:** Part IA Analysis I, Mathematical Tripos (Cambridge does not use numeric module codes)
- **Lectures:** 24 lectures
- **Term:** Lent Term, Year 1
- **Examined in:** Paper 1, with Vectors and Matrices
- **Prerequisites:** None listed in the schedules
- **Covers:** Sequences and series, convergence tests, continuity, differentiability, Rolle and the Mean Value Theorem, Taylor's theorem, power series, the Riemann integral, improper integrals
- **Practice here:** [Calculus 1](/learn/calculus-1) and the series units of [Calculus 2](/learn/calculus-2)

## Key takeaways

- Analysis I is a 24-lecture Lent Term course in Part IA of the Cambridge Mathematical Tripos, examined in Paper 1.
- It proves the foundations of single-variable calculus: limits, continuity, derivatives and the Riemann integral.
- It covers convergence tests for series and power series with radius of convergence.
- It ends with the Fundamental Theorem of Calculus, integration by parts, Taylor's theorem with integral remainder and improper integrals.

**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

| Schedule topic | Practice | Read |
| --- | --- | --- |
| Sequences and limits | [Sequences](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Series and partial sums | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series) |
| Comparison test | [Comparison tests](/learn/calculus-2/sequences-and-series/comparison-tests) | [Convergence tests](/blog/convergence-tests) |
| Ratio test | [Ratio and root tests](/learn/calculus-2/sequences-and-series/ratio-and-root-tests) | [Ratio test](/rules/ratio-test) |
| Alternating series test | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Continuity | [Continuity](/learn/calculus-1/limits/continuity) | [Continuity guide](/blog/continuity) |
| Intermediate Value Theorem | [Intermediate Value Theorem](/learn/calculus-1/limits/intermediate-value-theorem) | [IVT statement](/theorems/intermediate-value-theorem) |
| Bounded and attains its bounds | [Absolute extrema](/learn/calculus-1/applications-of-derivatives/absolute-extrema) | [Extreme Value Theorem](/theorems/extreme-value-theorem) |
| Differentiability and the chain rule | [Differentiability](/learn/calculus-1/the-derivative/differentiability) | [Chain rule](/rules/chain-rule) |
| Inverse functions | [Inverse function derivatives](/learn/calculus-1/derivative-rules/inverse-function-derivatives) | [Inverse function rule](/rules/inverse-function-rule) |
| Rolle and the Mean Value Theorem | [Linear approximation and the MVT](/learn/calculus-1/applications-of-derivatives/linear-approximation-and-mvt) | [Rolle's theorem](/theorems/rolles-theorem), [Mean Value Theorem](/theorems/mean-value-theorem) |
| Taylor's theorem | [Estimating with Taylor](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) | [Taylor's theorem](/theorems/taylors-theorem) |
| Power series | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series](/blog/power-series) |
| The Riemann integral | [Riemann sums](/learn/calculus-1/integrals/riemann-sums) | [Riemann sums guide](/blog/riemann-sums) |
| Fundamental Theorem of Calculus | [FTC part 1](/learn/calculus-1/integrals/fundamental-theorem-part-1) | [FTC part 1](/theorems/fundamental-theorem-of-calculus-part-1) |
| Integration by parts | [Integration by parts](/learn/calculus-2/integration-techniques/integration-by-parts) | [Integration by parts rule](/rules/integration-by-parts-rule) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals](/blog/improper-integrals) |

## Example problems at the Analysis I level

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

$$\frac{a_{n+1}}{a_n} = \frac{n+1}{2n} \to \frac{1}{2} < 1$$

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

$$f(b) - f(a) = f'(c)(b - a) = 0$$

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](/functions/absolute-value-function) are good places to start.
- **Keep computations automatic.** Fluent [convergence tests](/practice-problems/convergence-tests) and [Taylor series](/practice-problems/taylor-series) leave your attention free for the logic.
- **Review the [Mean Value Theorem guide](/blog/mean-value-theorem)** 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](/courses/university-of-cambridge-differential-equations).

## Frequently asked questions

### What is Analysis I at Cambridge?

It is the first rigorous analysis course in the Mathematical Tripos. According to the 2026-27 schedules it covers limits and convergence, continuity, differentiability, power series and integration, in 24 lectures in Lent Term.

### How does Analysis I relate to school calculus?

It proves results you already use, such as the chain rule, the Mean Value Theorem and the Fundamental Theorem of Calculus, from the definitions. The computations are familiar, but the emphasis is on careful proof.

### How is Analysis I examined?

In Paper 1 of Part IA, where each section is split equally between Analysis I and Vectors and Matrices. You may answer all of Section I and at most five Section II questions, with no more than three from one course.

### How can I prepare for Analysis I?

Get comfortable with sequences and series, and make limit and derivative computations automatic so you can focus on the logic of each proof. Numbers and Sets in Michaelmas also introduces least upper bounds and convergence of monotone sequences.