Infinite Sequences & Series

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 10. This unit is worth 15–20% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• Sequences, Geometric Series and Telescoping Sums15 min · 3 objectivesDistinguish a sequence from a series and the convergence of each · Decide whether a geometric series converges and find its sum when it does · Evaluate a telescoping series by examining its partial sums
• 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
• Choosing a Test, and Absolute versus Conditional Convergence16 min · 3 objectivesSelect an appropriate convergence test from the form of the terms · Distinguish absolute from conditional convergence and test for each in the right order · State a convergence conclusion with the hypotheses the test actually requires
• 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
• 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
• 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
• Building New Series from the Ones You Know16 min · 3 objectivesRecall the four required Maclaurin series and their intervals of convergence · Produce a new series by substitution, multiplication, differentiation or integration of a known one · Use a series to evaluate a limit or an integral that resists other methods

Formulas in Unit 10

Every term in Unit 10

All 63 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 lim|aₙ₊₁/aₙ| = L: converges absolutely if L < 1, diverges if L > 1, inconclusive if L = 1. The workhorse for radius of convergence.

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.

Test-selection order

nth-term, then geometric or p-series, then alternating, then ratio if factorials or nth powers, then limit comparison for rational terms.

nth-term test proves divergence only

If aₙ → 0 the test is inconclusive. "Terms go to zero, so it converges" is the costliest false sentence in the unit.

The harmonic counterexample

1/n → 0 and Σ1/n diverges. The standing refutation of the false converse.

Geometric series sum

Σarⁿ = a/(1 − r) for |r| < 1. The only common test that also gives the sum, which is why sums asked for are usually geometric.

First term matters for the sum

a is the first term actually present. A series starting at n = 2 has a different a from one starting at n = 0.

p-series threshold

Σ1/nᵖ converges exactly when p > 1, linked to the p-integral by the integral test.

Ratio test at L = 1

Inconclusive — and it is always 1 at the endpoints of a power series, which is why endpoints need a different test.

Ratio test is for factorials and nth powers

They cancel cleanly in the ratio. For a rational function of n the ratio tends to 1 and the test says nothing.

Limit comparison choice

Compare with the p-series matching the degrees: for (3n² + 1)/(n⁴ − 2) compare with 1/n².

Direct comparison direction

Smaller than a convergent series proves convergence; larger than a divergent series proves divergence. The other two combinations prove nothing.

Integral test conditions

f must be positive, continuous and decreasing on the interval. Skipping the check is a common rubric loss.

Absolute convergence implies convergence

If Σ|aₙ| converges then Σaₙ converges. The converse fails — the alternating harmonic series is the example.

Conditional convergence

Σaₙ converges while Σ|aₙ| diverges. The alternating harmonic series is the canonical case.

Alternating series test conditions

Terms must decrease in magnitude AND tend to zero. Both are required; either alone is insufficient.

Alternating error bound

The error after n terms is less than the magnitude of the first OMITTED term. No derivatives needed.

Lagrange bound: which n to use

Both the power and the factorial use n + 1, the degree of the first OMITTED term — not n. Using n is the most frequent error in applying the bound.

Which error bound to use

A visibly alternating series takes the simpler bound. A given bound on a higher derivative signals Lagrange.

Taylor polynomial construction

Each term needs a derivative value at the center, a power of (x − a), and a factorial. Dropping the factorial is the usual error.

Center matters

Powers of (x − a), not of x, whenever the center is not 0. A Maclaurin series is the special case a = 0.

Maclaurin for eˣ

Σxⁿ/n!, converging for all real x. Radius infinite, so there are no endpoints to test.

Maclaurin for sin x

x − x³/3! + x⁵/5! − …, odd powers only, alternating. Converges everywhere.

Maclaurin for cos x

1 − x²/2! + x⁴/4! − …, even powers only, alternating. Converges everywhere.

Maclaurin for 1/(1 − x)

Σxⁿ, the geometric series, converging for |x| < 1. The parent of most manipulation questions.

Sine and cosine skip degrees

P₃ and P₄ for sin x are the same polynomial, so the first omitted term after P₃ is the x⁵ term. Overlooking this makes an error bound far too loose.

Radius then endpoints

The ratio test gives R. The endpoints must then be substituted and tested separately, because the ratio test is inconclusive there.

Endpoints can disagree

Σxⁿ/n has interval [−1, 1): divergent harmonic at x = 1, convergent alternating harmonic at x = −1.

Radius zero or infinite

R = 0 means convergence only at the center; R = ∞ means convergence everywhere and no endpoints to check.

Substitution into a known series

For e^(x²) put x² wherever x appears: Σx²ⁿ/n!. Squaring the whole term would also square the factorial, which is different.

Radius survives term-by-term operations

Differentiating or integrating a power series term by term leaves R unchanged, but can change what happens AT the endpoints, so those must be retested.

Series for ln(1 + x)

Obtained by integrating the geometric series for 1/(1 + x) term by term. A standard manipulation rather than a memorized series.

Multiplying by a power of x

x·sin x shifts every exponent up by one. Radius is unchanged.

Using a series to evaluate a hard integral

Expand the integrand as a series and integrate term by term — the standard route for ∫sin(x²)dx and similar, which have no elementary antiderivative.

A Taylor polynomial is finite

A polynomial approximates near the center; the series is the infinite object. Questions about error concern the polynomial.

What examiners penalize here

• Writing "aₙ → 0, therefore the series converges" is scored as a wrong statement even when the series does happen to converge. The rubric penalizes the reasoning, not just the conclusion.
• 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.
• Free-response convergence questions want the test named and its hypotheses verified. "By the alternating series test, since 1/√(n+3) is decreasing and tends to 0" earns credit; "it converges by AST" alone often does not.
• 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!.
• 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.
• 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.
• Series free-response questions are usually four parts that all rest on one expansion: write the series, differentiate or integrate it, bound the error, then state the interval. Getting the first part right carries the rest, so spend the time 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