Infinite Sequences & Series

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 10. This unit is worth 17–18% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Series Convergence Tests16 min · 3 objectivesApply the nth-term test, geometric series test, and p-series test · Use the ratio test to determine convergence · Distinguish absolute from conditional convergence
• Taylor & Maclaurin Series16 min · 3 objectivesConstruct a Taylor polynomial centered at a given point · Recognize the Maclaurin series for eˣ, sin x, and cos x · Use known series to build new ones by substitution
• Power Series & Interval of Convergence15 min · 3 objectivesFind the radius of convergence using the ratio test · Determine the interval of convergence, checking the endpoints · Understand term-by-term differentiation and integration of power series
• Error Bounds14 min · 3 objectivesBound the error of an alternating series by the first omitted term · Apply the Lagrange error bound for a Taylor polynomial · Interpret an error bound as a guarantee on approximation accuracy

Formulas in Unit 10

Every term in Unit 10

All 30 terms we publish for Infinite Sequences & Series, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

Lagrange error bound

|Rₙ(x)| ≤ M|x − a|ⁿ⁺¹/(n + 1)!, where M bounds the (n+1)th derivative on the interval. State what M is and why.

Geometric series

Σarⁿ converges to a/(1 − r) when |r| < 1 and diverges otherwise. The only common series whose exact sum is easy to state.

p-series

Σ1/nᵖ converges for p > 1 and diverges for p ≤ 1. The harmonic series p = 1 diverges, which is the case worth memorizing.

Ratio test

Compute L = lim|aₙ₊₁/aₙ|. L < 1 converges absolutely, L > 1 diverges, L = 1 is inconclusive and a different test is needed.

Interval of convergence endpoints

The ratio test says nothing at the endpoints, so substitute each one and test the resulting numerical series separately.

nth term test for divergence

If the terms do not approach zero, the series diverges. It can NEVER prove convergence — terms approaching zero proves nothing.

Integral test

For positive, decreasing, continuous terms, the series and ∫f(x)dx converge together. The integral's value is not the series sum.

Direct comparison test

Requires positive terms. Smaller than a convergent series means convergent; larger than a divergent series means divergent. The other two combinations conclude nothing.

Limit comparison test

If lim(aₙ/bₙ) is a positive finite number, both series do the same thing. Easier than direct comparison because no inequality must be proved.

Alternating series test

Terms alternate, decrease in absolute value, and approach zero — then the series converges. It cannot establish absolute convergence.

Alternating series error bound

The error after n terms is less than the absolute value of the first omitted term. Applies only to alternating series meeting the test's conditions.

Absolute vs conditional convergence

Absolute means Σ|aₙ| converges. Conditional means Σaₙ converges but Σ|aₙ| does not — the alternating harmonic series is the standard example.

Radius and interval of convergence

Apply the ratio test and solve L < 1 for x. The endpoints must then be tested separately, since the ratio test is inconclusive there.

Power series centered at a

Σcₙ(x − a)ⁿ converges on an interval centered at a. Inside the radius it converges absolutely; outside it diverges.

Taylor series

f(x) = Σ f⁽ⁿ⁾(a)(x − a)ⁿ/n!. A Maclaurin series is the special case a = 0.

Maclaurin series to know

eˣ = Σxⁿ/n!, sin x = Σ(−1)ⁿx^(2n+1)/(2n+1)!, cos x = Σ(−1)ⁿx^(2n)/(2n)!, 1/(1 − x) = Σxⁿ for |x| < 1.

Building new series from known ones

Substitute, multiply by a power of x, differentiate or integrate a known series term by term — far faster than computing derivatives.

Taylor polynomial vs Taylor series

A polynomial is a finite truncation used to approximate; the series is the infinite sum, which equals the function only where it converges to it.

Term-by-term differentiation and integration

Valid inside the interval of convergence, and the radius is unchanged — though endpoint behavior can change.

Sequence vs series

A sequence is a list of terms; a series is their sum. A sequence can converge while its series diverges — 1/n is the standard case.

Telescoping series

Consecutive terms cancel, leaving a few. Write out several partial sums to see the pattern before taking the limit.

Partial sums

Sₙ is the sum of the first n terms. A series converges exactly when the sequence of partial sums converges.

Choosing a convergence test

Terms not approaching zero → nth term test. Factorials or powers of n → ratio test. Looks like 1/nᵖ → comparison or limit comparison. Alternating signs → alternating series test.

Root test

L = lim ⁿ√|aₙ|; same conclusions as the ratio test. Useful when the whole term is raised to the nth power.

Error bound for a Taylor polynomial

Lagrange: |Rₙ| ≤ M|x − a|ⁿ⁺¹/(n + 1)!. Identify M as a bound on the (n+1)th derivative and say why it bounds it.

Alternating series error bound in practice

The error is smaller than the first omitted term, so a required accuracy determines how many terms are needed.

Series for a definite integral

Substituting into a known Maclaurin series and integrating term by term evaluates integrals with no elementary antiderivative, such as ∫e^(−x²)dx.

Common series to recognize instantly

Σ1/n diverges, Σ1/n² converges, Σ(−1)ⁿ⁺¹/n converges conditionally, and Σ1/n! converges to e − 1 without the n = 0 term.

Manipulating the geometric series

From 1/(1 − x) = Σxⁿ, substituting −x² gives 1/(1 + x²), and integrating that gives the arctan series.

Taylor series centered elsewhere

Expanding about a ≠ 0 uses powers of (x − a) and derivatives evaluated at a. Convergence is fastest near the center.

What examiners penalize here

• Match the test to the terms: factorials or nth powers → ratio test; a clean 1/nᵖ → p-series; a constant ratio between terms → geometric. Recognizing the *form* quickly is the skill that makes the series free-response manageable.
• Coefficient and derivative are linked by a factorial: the coefficient of (x − a)ⁿ is fⁿ(a)/n!, so fⁿ(a) = n! × (that coefficient). AP loves asking for a high derivative at the center — read off the coefficient and multiply by n!.
• A complete interval-of-convergence answer has three parts: apply the ratio test for R, then test each endpoint separately, then state the interval with the correct brackets. Free-response rubrics award a point specifically for the endpoint analysis.
• Choose the bound that fits: if the series alternates with decreasing terms, use the simple "first omitted term" bound. Otherwise reach for Lagrange, and take M as the maximum of the (n+1)th derivative on the relevant interval — often just its largest possible value there.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series