# AP Calc BC Unit 10 Review: Infinite Sequences and Series

> AP Calculus BC Unit 10 review: convergence tests, error bounds, Taylor and Maclaurin series, and the radius and interval of convergence, with practice.

Source: https://practicecalculus.com/ap-calculus/unit-10-infinite-sequences-and-series
Updated: 2026-11-06

## At a glance

- **AB exam weight:** Not on the AB exam
- **BC exam weight:** 15–20%
- **Big ideas:** Convergence tests, Taylor polynomials, error bounds, power series
- **BC only:** The whole unit
- **Comes after:** Unit 9, parametric, polar and vector-valued functions
- **Practice here:** [Calculus 2: Sequences and series](/learn/calculus-2/sequences-and-series/series-and-partial-sums)

## Key takeaways

- Unit 10 is a BC-only unit worth 15 to 20 percent of the BC exam, tied for the largest share of any BC unit.
- A geometric series $\sum_{n=0}^{\infty} ar^n$ converges to $\frac{a}{1 - r}$ when $|r| < 1$ and diverges otherwise.
- The Taylor series of $f$ about $x = a$ is $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$.
- Two error bounds: the alternating series bound uses the first omitted term, and the Lagrange bound uses $\frac{M}{(n + 1)!}|x - a|^{n+1}$.

**AP Calculus Unit 10, Infinite Sequences and Series**, is the second BC-only unit and the largest. It asks a surprising question: can you add infinitely many numbers and get a finite answer? Sometimes you can. Then it uses that idea to write functions like $e^x$ and $\sin x$ as infinite polynomials, which is how calculators actually compute them.

| | AP Calculus AB | AP Calculus BC |
| --- | --- | --- |
| Exam weight | not on the AB exam | 15–20% (BC only) |
| Extra topics | none | the whole unit is BC only |

## What Unit 10 covers

College Board splits the unit into these topics. Each row links to a lesson where you can practice it with new numbers every time.

| Topic | What you learn | Practice |
| --- | --- | --- |
| Convergent and divergent series | Partial sums and what convergence means | [Series and partial sums](/learn/calculus-2/sequences-and-series/series-and-partial-sums) |
| Geometric series | $\sum ar^n = \frac{a}{1 - r}$ for $\lvert r\rvert < 1$ | [Geometric series](/learn/calculus-2/sequences-and-series/geometric-series) |
| The nth term test | If $a_n \not\to 0$, the series diverges | [nth term test](/learn/calculus-2/sequences-and-series/nth-term-test) |
| Integral test, harmonic series and p-series | $\sum \frac{1}{n^p}$ converges only when $p > 1$ | [Integral test and p-series](/learn/calculus-2/sequences-and-series/integral-test-and-p-series) |
| Comparison tests | Direct and limit comparison | [Comparison tests](/learn/calculus-2/sequences-and-series/comparison-tests) |
| Alternating series test | Decreasing terms that go to 0 | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) |
| Ratio test | Using $\lim \left\lvert\frac{a_{n+1}}{a_n}\right\rvert$ | [Ratio and root tests](/learn/calculus-2/sequences-and-series/ratio-and-root-tests) |
| Absolute and conditional convergence | Whether $\sum \lvert a_n\rvert$ also converges | [Choosing a test](/learn/calculus-2/sequences-and-series/choosing-a-test) |
| Alternating series error bound | Error is at most the first omitted term | [Alternating series](/learn/calculus-2/sequences-and-series/alternating-series) |
| Taylor polynomials | Approximating a function near a point | [Taylor polynomials](/learn/calculus-2/power-series-parametric-polar/taylor-polynomials) |
| Lagrange error bound | Bounding a Taylor approximation's error | [Estimating with Taylor](/learn/calculus-2/power-series-parametric-polar/estimating-with-taylor) |
| Radius and interval of convergence | The ratio test plus checking endpoints | [Power series and radius](/learn/calculus-2/power-series-parametric-polar/power-series-and-radius) |
| Taylor and Maclaurin series | Building series for $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1 - x}$ | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) |
| Representing functions as power series | Substituting, differentiating and integrating series | [Maclaurin series](/learn/calculus-2/power-series-parametric-polar/maclaurin-series) |

## The key ideas

**A series converges when its partial sums settle down.** If $S_N = a_1 + a_2 + \dots + a_N$ approaches a finite limit, the series converges to that limit. The terms going to 0 is necessary but not enough: the harmonic series $\sum \frac{1}{n}$ has terms that go to 0, yet it diverges.

**Each test fits a shape.** The [nth term test](/rules/nth-term-test) only proves divergence. The [geometric series formula](/rules/geometric-series-formula) gives an exact sum. The [ratio test](/rules/ratio-test) handles factorials and exponentials. The [alternating series test](/rules/alternating-series-test) needs terms that decrease to 0.

**Taylor polynomials match a function near a point.** The degree $n$ Taylor polynomial matches $f$ and its first $n$ derivatives at $x = a$. More terms give a better fit over a wider interval.

```graph
title: 'cos x and its degree 4 Maclaurin polynomial'
caption: 'The polynomial $1 - \frac{x^2}{2} + \frac{x^4}{24}$ is almost exactly $\cos x$ near 0 and drifts away as $|x|$ grows.'
x: [-5, 5]
y: [-2, 3]
functions:
  - expr: cos(x)
    label: y = cos x
  - expr: 1 - x^2/2 + x^4/24
    label: P₄(x) = 1 − x²/2 + x⁴/24
    dashed: true
```

**Power series converge on an interval.** Use the ratio test to find the radius $R$, then test each endpoint separately, since the ratio test says nothing there. Read more in the guides on [convergence tests](/blog/convergence-tests), [power series](/blog/power-series) and [Taylor series](/blog/taylor-series), and the [Taylor's Theorem](/theorems/taylors-theorem) page.

## Sample questions

**1. Geometric series.** $\sum_{n=0}^{\infty} 3\left(\frac{1}{2}\right)^n$ has $a = 3$ and $r = \frac{1}{2}$, so it converges to $\frac{3}{1 - 1/2} = 6$.

**2. Ratio test.** For $\sum_{n=1}^{\infty} \frac{n}{3^n}$:

$$\lim_{n \to \infty} \frac{n + 1}{3^{n+1}}\cdot\frac{3^n}{n} = \lim_{n \to \infty} \frac{n + 1}{3n} = \frac{1}{3} < 1$$

so the series converges.

**3. Alternating series error.** The alternating harmonic series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ converges to $\ln 2$. The first four terms give $\frac{7}{12} \approx 0.583$, and the error is at most the next term, $\frac{1}{5}$. The actual error is about $0.110$.

**4. Interval of convergence.** For $\sum_{n=1}^{\infty} \frac{x^n}{n}$, the ratio test gives $|x| < 1$. At $x = 1$ it is the harmonic series, which diverges. At $x = -1$ it is the alternating harmonic series, which converges. The interval is $[-1, 1)$.

**5. Lagrange error bound.** The degree 2 Maclaurin polynomial of $e^x$ gives $e^{0.1} \approx 1 + 0.1 + 0.005 = 1.105$. Every derivative of $e^x$ is $e^x$, which is less than 3 on $[0, 0.1]$, so

$$\text{error} \le \frac{3}{3!}(0.1)^3 = 0.0005$$

The true value is $e^{0.1} \approx 1.105171$, so the error is about $0.00017$.

## How to study Unit 10

- **Make a test flowchart.** Decide in what order you check nth term, geometric, p-series, alternating, ratio and comparison, then practice choosing quickly.
- **Always test the endpoints.** The interval of convergence free-response question gives a point for each endpoint, and they often behave differently.
- **Memorize four Maclaurin series.** $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1 - x}$ cover most of the exam once you can substitute into them.
- **Write the error bound in full.** Show $M$, the factorial and the power, not just a number.
- **Work through problem sets.** Try the [convergence tests practice problems](/practice-problems/convergence-tests), [power series problems](/practice-problems/power-series) and [Taylor series problems](/practice-problems/taylor-series), each with full solutions.

For the full course map, see the [AP Calculus BC guide](/ap-calculus/bc), or compare the two courses in the [AP Calculus AB guide](/ap-calculus/ab).

> AP and Advanced Placement are registered trademarks of College Board, which is not affiliated with PracticeCalculus and does not endorse this guide. Unit names and weights come from College Board's course framework; check it for any changes.

## Frequently asked questions

### What is AP Calculus Unit 10 about?

Unit 10 asks when an infinite sum adds up to a finite number. You learn the geometric, nth term, integral, p-series, comparison, alternating series and ratio tests, then use Taylor polynomials and power series to approximate functions and bound the error.

### Is Unit 10 on the AP Calculus AB exam?

No. Unit 10 is only in AP Calculus BC, where it is 15 to 20 percent of the multiple-choice score. There is also almost always a full free-response question on Taylor series and the interval of convergence.

### Which convergence test should I use?

Start with the nth term test to rule out divergence. Then look at the form: geometric and p-series have their own rules, alternating signs suggest the alternating series test, and factorials or powers like $3^n$ suggest the ratio test. Comparison tests work for terms that look like a known series.

### Which Maclaurin series should I memorize?

Memorize the series for $e^x$, $\sin x$, $\cos x$ and $\frac{1}{1 - x}$. Many exam questions build new series from these by substituting, differentiating or integrating term by term.