AP Calculus BC only

Unit 10: Infinite Sequences and Series

Exam weighting: AB n/a · BC 15-20%

Unit 10, BC-only, covers infinite series: deciding whether a sum of infinitely many terms settles on a finite value, and for power series, which function it represents. You learn convergence tests, two error bounds, and Taylor and Maclaurin series. It is 17-18% of the BC exam, tied with Unit 6 for the heaviest weight.

The core idea

An infinite series adds up infinitely many terms, and the whole unit turns on one question: does that running total settle on a finite number? Topics 10.1 through 10.10 answer it for a series of constants, and Topics 10.11 through 10.15 go further, letting a series of powers of xx stand in for an entire function. The skill the exam rewards, and a solver cannot fake, is looking at a series and knowing which test to reach for.

The unit runs in two halves. Topic 10.1 defines convergence through the sequence of partial sums: a series converges exactly when that sequence has a limit. Topics 10.2 through 10.9 build the toolkit for deciding it: geometric series, the nnth term test, the integral test, harmonic and p-series, comparison and limit comparison, the alternating series test, and the ratio test, ending with absolute versus conditional convergence. Topic 10.10 bounds the error of an alternating partial sum. The second half (Topics 10.11 through 10.15) uses these ideas to represent functions with Taylor and Maclaurin polynomials and series, the Lagrange error bound, and the radius and interval of convergence of a power series.

Series shapeTest (topic)Converges when
Terms do not shrink to 00nnth term test (10.3)Never: it diverges
Constant ratio, ∑arn\sum ar^nGeometric (10.2)∣r∣<1|r| < 1, sum a1−r\frac{a}{1-r}
Terms like 1np\frac{1}{n^p}p-series (10.5)p>1p > 1
Looks like a series you knowComparison, limit comparison (10.6)It beats or matches that series
Positive and easy to integrateIntegral test (10.4)Its improper integral converges
Alternating, terms decreasing to 00Alternating series test (10.7)Terms decrease to 00
Factorials or nnth powersRatio test (10.8)L<1L < 1
lim⁡n→∞∣an+1an∣=L  ⇒  {L<1convergesL>1divergesL=1inconclusive\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right| = L \;\Rightarrow\; \begin{cases} L < 1 & \text{converges} \\ L > 1 & \text{diverges} \\ L = 1 & \text{inconclusive} \end{cases}

Once a series converges, Topic 10.9 asks what kind. It converges absolutely if the series of absolute values ∑∣an∣\sum |a_n| also converges, and absolute convergence guarantees the original converges. A series that converges while ∑∣an∣\sum |a_n| diverges converges conditionally: the alternating harmonic series ∑(−1)n+1n\sum \frac{(-1)^{n+1}}{n} is the standard example, since it converges but ∑1n\sum \frac{1}{n} does not. On the exam, test ∑∣an∣\sum |a_n| first, then fall back to the alternating series test if that diverges.

These topics turn a function into an infinite polynomial. The Taylor polynomial centered at x=ax = a is built from the derivatives of ff at aa, with the nnth-degree coefficient f(n)(a)n!\frac{f^{(n)}(a)}{n!} (Topic 10.11); keep every term and you have the Taylor series, or the Maclaurin series when a=0a = 0. Memorize the Maclaurin series for exe^x, sin⁡x\sin x, cos⁡x\cos x, and 11−x\frac{1}{1-x} (Topic 10.14), then build new series from them by substitution, term-by-term differentiation, or term-by-term integration (Topic 10.15). A power series converges only on an interval: the ratio test gives the radius of convergence, and you test each endpoint separately to finish the interval (Topic 10.13).

The two error bounds

Two question types ask how far a partial sum or Taylor polynomial sits from the true value, and each has its own bound. Read the series first, then pick the matching bound. The alternating series error bound (Topic 10.10) applies only to a convergent alternating series: the error is at most the size of the first term you drop, ∣S−Sn∣≤∣an+1∣|S - S_n| \le |a_{n+1}|. The Lagrange error bound (Topic 10.12) works for any Taylor polynomial, capping the remainder:

∣Rn(x)∣≤max⁡∣f(n+1)(z)∣(n+1)! ∣x−a∣n+1|R_n(x)| \le \frac{\max |f^{(n+1)}(z)|}{(n+1)!}\,|x - a|^{n+1}

What the exam asks

Unit 10 is 17-18% of the BC exam, tied with Unit 6 for the heaviest weight, and it is tested only on BC. One line from the CED settles your study list: the nnth term test, integral test, comparison and limit comparison tests, alternating series test, and ratio test are the only convergence tests the exam assesses (Topics 10.3-10.8), so a root test or others are not needed. Multiple-choice questions lean on test selection and geometric-series sums; free-response questions build a Taylor or Maclaurin series, find its interval of convergence, or bound an error. Progress Check 10 has about 45 multiple-choice questions and 3 free-response questions.

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 10.1Defining Convergent and Divergent Infinite Series
  • 10.2Working with Geometric Series
  • 10.3The nth Term Test for Divergence
  • 10.4Integral Test for Convergence
  • 10.5Harmonic Series and p-Series
  • 10.6Comparison Tests for Convergence
  • 10.7Alternating Series Test for Convergence
  • 10.8Ratio Test for Convergence
  • 10.9Determining Absolute or Conditional Convergence
  • 10.10Alternating Series Error Bound
  • 10.11Finding Taylor Polynomial Approximations of Functions
  • 10.12Lagrange Error Bound
  • 10.13Radius and Interval of Convergence of Power Series
  • 10.14Finding Taylor or Maclaurin Series for a Function
  • 10.15Representing Functions as Power Series

How to study this unit

  • Build a convergence-test reflex: run the $n$th term test first (Topic 10.3), then check whether the series is geometric (Topic 10.2) or a p-series (Topic 10.5) you can classify on sight, and save the ratio test (Topic 10.8) for terms with factorials or $n$th powers.
  • Remember the $n$th term test (Topic 10.3) can only prove divergence. If $\lim_{n \to \infty} a_n = 0$ the series may still diverge: the harmonic series $\sum \frac{1}{n}$ (Topic 10.5) is the classic trap where terms go to $0$ but the sum grows without bound.
  • Memorize the four core Maclaurin series, $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$ (Topic 10.14), then generate everything else by substituting, differentiating, or integrating term by term (Topic 10.15) instead of recomputing derivatives from scratch.
  • For an interval of convergence (Topic 10.13), use the ratio test to get the radius, then test each endpoint separately with a different test. The ratio test is always inconclusive at the endpoints, so a series can converge at one, both, or neither.
  • Match the error bound to the series: the alternating series error bound (Topic 10.10) is just the size of the first dropped term and only works for alternating series, while the Lagrange error bound (Topic 10.12) needs the max of the next derivative and works for any Taylor polynomial.

Guides for this unit

Tools and tables for this unit