AP Calculus BC series, made clear

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

Infinite series is the most abstract part of AP Calculus BC. Here is a clear map of the ideas and the tests you need.

Sequences and series

A sequence is a list of numbers. A series is the sum of a sequence. The key question is whether an infinite series adds up to a finite value (converges) or not (diverges).

The convergence tests

• nth term test: if the terms do not approach 0, the series diverges. If they do, you learn nothing yet.
• Geometric series: Σarⁿ converges when |r| < 1, to a/(1 − r).
• p-series: Σ1/n^p converges when p > 1.
• Integral test: compare the series with an improper integral.
• Comparison and limit comparison tests: compare with a series you already know.
• Alternating series test: an alternating series converges if its terms decrease in size towards 0.
• Ratio test: useful for factorials and powers; converges if the limit of |a_n+1/a_n| is less than 1.

Absolute and conditional convergence

A series converges absolutely if the series of absolute values converges. It converges conditionally if it converges but the absolute values do not. The alternating harmonic series is the classic example.

Taylor and Maclaurin series

A Taylor polynomial approximates a function near a point using its derivatives. A Maclaurin series is a Taylor series centred at 0. Know these by heart:

• e^x = 1 + x + x²/2! + x³/3! + …
• sin x = x − x³/3! + x⁵/5! − …
• cos x = 1 − x²/2! + x⁴/4! − …
• 1/(1 − x) = 1 + x + x² + x³ + … for |x| < 1

Radius and interval of convergence

Use the ratio test to find the radius of convergence, then check each endpoint separately. Endpoint checks are a common free-response task.

Error bounds

The alternating series error bound says the error is at most the size of the first unused term. The Lagrange error bound gives a limit on the error of a Taylor polynomial using the next derivative.

How to study series

1. Make a decision chart: which test to try for which kind of series.
2. Practise a few series every week from the moment the unit starts.
3. Write the test you are using and show its conditions in every answer.

Worked examples

Example 1: the nth term test

Does Σ n/(n + 1) converge?

• The terms approach 1, not 0, so the series diverges by the nth term test.

Example 2: a p-series

Does Σ 1/n³ converge?

• p = 3 > 1, so it converges.

Example 3: conditional convergence

Does Σ (−1)ⁿ⁺¹/n converge?

• The terms decrease to 0, so it converges by the alternating series test.
• The absolute values give the harmonic series, which diverges, so the series converges conditionally.

Example 4: the ratio test

Does Σ n/2ⁿ converge?

• |a_n+1/a_n| = (n + 1)/(2n) → ½ < 1, so it converges.

Example 5: a Taylor polynomial

Find the third-degree Maclaurin polynomial for ln(1 + x).

• f(0) = 0, f'(0) = 1, f''(0) = −1, f'''(0) = 2.
• P₃(x) = x − x²/2 + x³/3.

Example 6: interval of convergence

Find the interval of convergence of Σ xⁿ/n.

• Ratio test: |x| < 1 gives a radius of 1.
• At x = 1: the harmonic series diverges. At x = −1: the alternating harmonic series converges.
• Interval: [−1, 1).

Example 7: alternating series error bound

Estimate Σ (−1)ⁿ⁺¹/n² using three terms, and bound the error.

• 1 − 1/4 + 1/9 ≈ 0.861.
• The error is at most the next term: 1/16 = 0.0625.

Practice questions

Try each one before opening the answer.

Key takeaways

• Always start with the nth term test: terms not tending to 0 means divergence.
• Know the geometric and p-series rules by heart.
• Use the ratio test for factorials and powers.
• Check endpoints separately for intervals of convergence.
• Memorise the Maclaurin series for e^x, sin x, cos x and 1/(1 − x).

Understanding series

Partial sums

An infinite series is defined by its partial sums: add the first term, then the first two, then the first three, and so on. If these partial sums approach a fixed number, the series converges to that number. If they grow without bound or keep jumping around, it diverges. Every convergence test is a shortcut for deciding what the partial sums do.

Why the harmonic series diverges

The terms 1, ½, ⅓, ¼, … shrink to zero, yet the sum grows forever. Group the terms: ⅓ + ¼ > ½, then ⅕ to ⅛ > ½, and so on. You can always find another group worth more than ½, so the sum is unbounded. This is why terms approaching zero is necessary but not enough for convergence.

Taylor series as infinite polynomials

A Taylor polynomial matches a function's value and derivatives at one point. The more terms you use, the better the approximation near that point. Taylor series let calculators find values of functions like sin x using only addition and multiplication.

A test-choosing strategy

1. Do the terms go to 0? If not, the series diverges.
2. Is it geometric or a p-series? Use those rules directly.
3. Are there factorials or nth powers? Try the ratio test.
4. Does it alternate in sign? Try the alternating series test.
5. Does it look like a known series? Compare it with one.

Frequently asked questions

Do I need to memorise all the tests?

Yes, and their conditions. Free-response questions ask you to name the test and show its conditions are met.

How are series examined?

Often as one free-response question: write a Taylor polynomial, find its interval of convergence and bound the error.

One more worked example: error with a Taylor polynomial

Use the second-degree Maclaurin polynomial for cos x to estimate cos(0.2), and bound the error.

• cos x ≈ 1 − x²/2, so cos(0.2) ≈ 1 − 0.02 = 0.98.
• The series alternates and its terms decrease, so the error is at most the next term: x⁴/24 = 0.0016/24 ≈ 0.000067.
• The true value is 0.980067…, well within the bound.

Study tip

Make a one-page chart of every convergence test with its conditions and a typical example. Rewrite it from memory until you can do it in five minutes.

Where to go next

Series questions often sit alongside parametric and polar questions in the BC exam. Review the rest of the BC course and the results you need to memorise.

• AP Calculus BC units, explained
• What to memorise for AP Calculus
• AP Calculus AB vs BC