# Stanford Math 21 Calculus: Series, Study Guide and Practice

> A study guide for Stanford Math 21: sequences, improper integrals, convergence tests, power series and Taylor approximations, mapped to free practice.

Source: https://practicecalculus.com/courses/stanford-math-21
Updated: 2026-11-03

## At a glance

- **Course code:** MATH 21
- **Units:** 4
- **Offered:** Autumn, winter and spring quarters (2026-27)
- **Covers:** Limit rules, sequences of numbers and functions, limits at infinity, integrals of rational functions, improper integrals, infinite series and convergence tests, power series and Taylor approximations
- **Previous course:** MATH 20 (Calculus)
- **Next course:** MATH 51 (Linear Algebra, Multivariable Calculus, and Modern Applications), which requires Math 21
- **Practice here:** [Calculus 2](/learn/calculus-2), improper integrals and series units

## Key takeaways

- Math 21 completes Stanford's single-variable series and is built around calculus with infinite processes.
- Its content is largely the BC-level AP Calculus material that is not in the AB syllabus.
- Topics run from sequences and improper integrals to series, power series and Taylor approximations.
- Math 21 is required to enroll in Math 51 and CME 100.

**Stanford Math 21** is the last course in Stanford's single-variable calculus series, after [Math 19](/courses/stanford-math-19) and [Math 20](/courses/stanford-math-20). It is the gateway to the multivariable courses: Math 51 requires it. The whole course is organized around one theme, infinite processes. This guide maps each Math 21 topic to free practice with instant feedback.

## What Stanford Math 21 covers

Stanford's catalog frames Math 21 as calculus with infinite processes, covering largely the BC-level AP Calculus content that is not in the AB syllabus. After a review of limit rules, it studies sequences of numbers and of functions and their limits at infinity. Integration continues with rational functions, picked up from Math 20, and especially improper integrals over unbounded intervals or of unbounded functions. The course then turns to infinite series, tests for convergence and divergence, and finally power series and Taylor approximations. The catalog motivates these with examples from probability, economics and computer science.

That is the series half of a typical **Calculus 2** course, plus improper integrals and partial fractions. Parametric curves and the start of integration techniques were already handled in Math 20.

## Math 21 topic map

| Math 21 topic | Practice | Read |
| --- | --- | --- |
| Limit rules review | [Limits at infinity](/learn/calculus-1/limits/limits-at-infinity) | [Limits at infinity guide](/blog/limits-at-infinity), [L'Hôpital's rule](/rules/lhopitals-rule) |
| Sequences | [Sequences and their limits](/learn/calculus-2/sequences-and-series/sequences) | [Squeeze theorem](/theorems/squeeze-theorem) |
| Rational functions and partial fractions | [Partial fractions](/learn/calculus-2/integration-techniques/partial-fractions) | [Partial fractions guide](/blog/partial-fractions) |
| Improper integrals | [Integrating to infinity](/learn/calculus-2/improper-integrals-and-applications/infinite-limits-of-integration) | [Improper integrals guide](/blog/improper-integrals) |
| Unbounded integrands | [Integrands that blow up](/learn/calculus-2/improper-integrals-and-applications/infinite-discontinuities) | [Improper integrals guide](/blog/improper-integrals) |
| Series and partial sums | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) | [Geometric series](/blog/geometric-series) |
| Convergence and divergence tests | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) | [Convergence tests](/blog/convergence-tests), [ratio test](/rules/ratio-test) |
| Alternating series | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) | [Alternating series test](/rules/alternating-series-test) |
| Power series | [Radius of convergence](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) | [Power series guide](/blog/power-series) |
| Taylor approximations | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) | [Taylor series](/blog/taylor-series), [Taylor's theorem](/theorems/taylors-theorem) |

## From Math 21 to the Math 50s

| Course | Units | Focus |
| --- | --- | --- |
| Math 21 | 4 | Single-variable calculus with infinite processes |
| Math 51 | 5 | Linear algebra and multivariable differential calculus |
| Math 52 | 5 | Integral calculus of several variables |
| Math 53 | 5 | Differential equations with linear algebra and Fourier methods |

Math 21 is required for [Math 51](/courses/stanford-math-51), and Math 52 and 53 both list Math 21 and Math 51 as prerequisites. See the [Math 52 guide](/courses/stanford-math-52) for multivariable integration.

## Example problems at the Math 21 level

**1. A limit of a sequence.**

$$\lim_{n\to\infty}\frac{3n^2 + 1}{n^2 - 5n} = \lim_{n\to\infty}\frac{3 + \frac{1}{n^2}}{1 - \frac{5}{n}} = 3$$

**2. The ratio test.** For $\sum \frac{n}{2^n}$:

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

so the series converges.

**3. Radius of convergence.** For $\sum \frac{x^n}{3^n}$, the ratio test gives $\frac{|x|}{3} < 1$, so the radius of convergence is $R = 3$.

## How to do well in Stanford Math 21

- **Know the two benchmark families.** Most comparisons go back to geometric series and p-series, so learn the [geometric series formula](/rules/geometric-series-formula) and practice the [integral test and p-series](/learn/calculus-2/sequences-and-series/integral-test-and-p-series).
- **Have a test-choosing routine.** Start with the [nth-term test](/rules/nth-term-test), then look at the form of the terms before reaching for the ratio test.
- **Practice Taylor series by building them,** not memorizing them. The [common Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) lesson shows how the main ones come from one another.
- **Review with the [formula sheet](/formulas)** and timed [practice exams](/practice), and read the [common calculus mistakes](/blog/common-calculus-mistakes) before the final.

## Frequently asked questions

### What is Stanford Math 21?

Math 21 is the third quarter of single-variable calculus at Stanford. Its theme is calculus with infinite processes: sequences, limits at infinity, improper integrals, infinite series and their convergence tests, and power series with Taylor approximations.

### What comes after Math 21?

Math 51, Linear Algebra, Multivariable Calculus, and Modern Applications, for which Math 21 is a requirement. Some students take CME 100 instead, which also requires Math 21.

### Does AP Calculus BC replace Math 21?

Stanford says the course is largely the BC-level material that is not in the AB syllabus. Whether your AP score gives you credit is set by Stanford's placement policy, so check the math department's placement page.

### Is Math 21 hard?

Series are often the least intuitive part of single-variable calculus, because you reason about whether an infinite process settles down. Steady practice with many short convergence problems is the most reliable fix.

### How should I prepare for Math 21?

Review limits, L'Hôpital's rule and integration by parts from Math 20, since improper integrals and convergence tests rely on all three.