AP Calculus BC · Exam guide
AP Calculus BC Series FRQ Explained: Taylor Polynomials, Error Bounds, and Convergence
The last free-response question on the AP Calculus BC exam is typically about series. It is a good place to earn points, because the parts follow a predictable pattern. This guide walks through the four moves that appear most often, with checked examples.
What the question usually asks
| Move | What you do |
|---|---|
| Build a Taylor polynomial | Use given values of the function and its derivatives at a point to write the polynomial of a stated degree. |
| Find the interval of convergence | Use the ratio test to find the radius, then check each endpoint on its own. |
| Bound an error | Use the Lagrange error bound or the alternating series error bound, and say which one applies. |
| Build a new series | Substitute into, differentiate, or integrate a known series, such as \(e^x\), \(\sin x\), \(\cos x\), or \(\dfrac{1}{1-x}\). |
Move 1: a Taylor polynomial
A function \(f\) has \(f(0) = 2\), \(f'(0) = 3\), \(f''(0) = -4\), and \(f'''(0) = 6\). (a) Write the third-degree Taylor polynomial for \(f\) about \(x = 0\). (b) Use it to approximate \(f(0.1)\).
The coefficients are \(f^{(n)}(0)/n!\): \(P_3(x) = 2 + 3x + \dfrac{-4}{2!}x^2 + \dfrac{6}{3!}x^3 = 2 + 3x - 2x^2 + x^3\). Then \(P_3(0.1) = 2 + 0.3 - 0.02 + 0.001 = 2.281\).
It uses the derivative values as coefficients without dividing by n!.
It shows the factorials, which makes the coefficients easy to check.
Move 2: the interval of convergence
Find the interval of convergence of \(\displaystyle\sum_{n=1}^{\infty}\dfrac{(x-1)^n}{n\,2^n}\).
By the ratio test, \(\left|\dfrac{a_{n+1}}{a_n}\right| \to \dfrac{|x-1|}{2}\), which is less than 1 when \(-1 \lt x \lt 3\). At \(x = 3\) the series is \(\sum \dfrac{1}{n}\), the harmonic series, which diverges. At \(x = -1\) it is \(\sum \dfrac{(-1)^n}{n}\), which converges by the alternating series test. The interval of convergence is \([-1, 3)\).
Graders give separate points for the radius and for each endpoint, so name the test you use at each endpoint. Skipping an endpoint is one of the most common ways to lose a point.
Move 3: an error bound
Suppose \(|f^{(4)}(x)| \le 24\) for all \(x\) in \([0, 0.1]\). Bound the error in approximating \(f(0.1)\) with \(P_3(0.1)\).
By the Lagrange error bound, \(|R_3(0.1)| \le \dfrac{M}{4!}\,|0.1|^4 = \dfrac{24}{24}(0.0001) = 0.0001\).
For an alternating series whose terms decrease in absolute value to 0, the alternating series error bound says the error is less than the first omitted term. State that the conditions hold before you use it.
Common mistakes
- Forgetting to divide by n! when building a Taylor polynomial.
- Stating the interval of convergence without testing the endpoints.
- Using the Lagrange error bound without a bound on the next derivative, or using the wrong derivative order.
- Using the alternating series error bound on a series that is not alternating or whose terms do not decrease.
Frequently asked questions
Is series always the last question on the AP Calculus BC exam?
It is typically the last free-response question in recent years, but check the current course and exam description.
Do I have to check endpoints?
Yes. The ratio test gives the radius, and the endpoints need separate tests.
Which error bound should I use?
Use the alternating series error bound for an alternating series with decreasing terms. Use the Lagrange error bound when you are given a bound on a higher derivative.
Can I use series I know, like the series for e to the x?
Yes. Building a new series from a known one is usually faster and is expected.
Does AP Calculus AB test series?
No. Series are BC only.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus BC course Free lessons and lesson list
Other free-response guides:
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