How many points is a series FRQ usually worth?

Series free-response questions are typically worth 9 points like other FRQs, split across several parts. Because parts are graded independently and reuse a common series, you can earn substantial credit even after an early mistake — so attempt every part.

When do I use the alternating series bound versus the Lagrange error bound?

Use the alternating series error bound when the series alternates in sign with terms decreasing to zero; the error is at most the first omitted term. Use the Lagrange error bound for general Taylor approximations, especially when the series is not alternating or the problem gives you a bound on a higher derivative.

Do I always have to check both endpoints for interval of convergence?

Yes. The Ratio Test gives you the open interval, but it is inconclusive at the endpoints. You must substitute each endpoint into the series and apply a separate test (alternating series, p-series, or the nth-term test). Skipping endpoint checks is the most common lost point.

Can I compute derivatives directly instead of manipulating a known series?

You can, but it is slower and error-prone. For functions like x^2 e^-x, (x^2), or x, start from a memorized Maclaurin series and substitute, multiply, differentiate, or integrate. This is faster and matches what graders expect for full credit.

AP-CALCBC-10-FRQ

U10 FRQ Practice

Master AP Calculus BC Unit 10 free-response questions: Taylor and Maclaurin series, convergence tests, error bounds, and interval of convergence, with a full worked FRQ.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on U10 FRQ Practice, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start U10 FRQ Practice free

What this lesson covers

Unit 10 free-response questions are among the most predictable and point-rich on the AP Calculus BC exam. A single series FRQ often bundles together everything you learned this unit: writing a Taylor polynomial, bounding error with the Lagrange or alternating series estimate, finding a radius and interval of convergence, and manipulating a known Maclaurin series to build a new one.

This guide will not re-teach each test — it assumes you know them. Instead it shows you how to attack the FRQ format: how graders assign points, which phrases earn credit, and how to avoid the small algebra and justification slips that cost easy points. By the end you will have a repeatable playbook for the classic "here is a Taylor series, now do five things with it" problem.

Anatomy of a Unit 10 FRQ

Series FRQs are built from modular parts, and each part is graded independently. If you blank on part (a), you can still earn parts (c) and (d). The most common building blocks appear below.

Task	What graders want	Typical points
Write first several terms / general term	Correct coefficients AND a general $n$ th term	1–2
Taylor polynomial about $x=a$	Uses $\frac{f^{(n)}(a)}{n!}(x-a)^n$ correctly	1–2
Radius/interval of convergence	Ratio Test setup + endpoint checks	2–3
Error bound	Names the bound (Lagrange or alternating) and evaluates	1–2
Manipulate a known series	Substitute, differentiate, integrate, multiply	1–2

A crucial habit: read all parts before starting. Later parts frequently reuse the series from part (a), so a mistake early can propagate. Write your general term clearly — graders look for it explicitly. When a problem gives you

f^{(n)}(a)

values in a table, plug them into the Taylor formula directly rather than trying to guess a pattern.

Building and using Taylor polynomials

The Taylor polynomial of degree

n

for

f

centered at

x=a

P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k.

Most FRQs either give you derivative values at

a

or ask you to differentiate a function a few times. The single most common error is forgetting the

k!

in the denominator, or mismatching a derivative value with the wrong power of

(x-a)

.

When the center is

x=0

you get a Maclaurin polynomial, and you should have the standard series memorized:

e^x

\sin x

\cos x

\frac{1}{1-x}

, and

\ln(1+x)

. Graders reward using these directly. If asked for the Taylor series of

x^2 e^{-x}

, do NOT compute five derivatives by hand — start from

e^u=\sum \frac{u^k}{k!}

, substitute

u=-x

, then multiply every term by

x^2

.

When approximating a function value, plug the specific

x

into your polynomial and simplify to a number if asked. Show the substituted terms before combining; a numeric-only answer with no work risks losing the point if the value is slightly off.

Convergence and error bound justification

Two parts appear almost every year: finding the interval of convergence and bounding approximation error. Both require explicit justification language.

For interval of convergence, apply the Ratio Test: compute

\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=L

, set

L<1

to get the radius, then TEST BOTH ENDPOINTS separately by substituting them into the series and applying an appropriate test (often the alternating series test or a

p

-series comparison). Missing endpoint checks is the number-one lost point here.

For error, identify which bound applies.

Situation	Bound to use	Form
Alternating series, terms decreasing to 0	Alternating Series Error	$\|error\|\le$ first omitted term
General Taylor approximation	Lagrange Error Bound	$\|R_n(x)\|\le \frac{\max\|f^{(n+1)}(c)\|}{(n+1)!}\|x-a\|^{n+1}$

For Lagrange, state the max value of the

(n+1)

th derivative on the interval between

a

and

x

; if the derivative is bounded (e.g. sines and cosines by 1), say so. Always write the inequality, then compute the number. Graders want to see the bound expression, not just a final decimal.

Manipulating known series efficiently

Series FRQs love the phrase "use your answer to part (a) to find..." This tests the four legal manipulations: substitution, term-by-term differentiation, term-by-term integration, and multiplication by a power of

x

.

If you have the series for

f(x)

and need

f'(x)

, differentiate each term. If you need

\int f(x)\,dx

, integrate each term and remember the constant

+C

, which you find using a known value like

f(0)

. Integration is common for functions like

\arctan x

(integrate

\frac{1}{1+x^2}

) or

\ln(1+x)

(integrate

\frac{1}{1+x}

).

A frequent setup: "Write the Maclaurin series for

g(x)=\int_0^x f(t)\,dt

." Take the series for

f

, integrate each power of

t

, and evaluate from

0

x

. The radius of convergence is preserved under differentiation and integration (though endpoint behavior can change).

Watch signs carefully when substituting negatives, and keep the general term with the summation symbol so the grader sees the pattern. A general term written as

\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}

earns more reliably than three loose terms and dots.

Key terms

Taylor polynomial.: The degree- $n$ polynomial approximating $f$ near $x=a$ , given by $\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k$ .
Maclaurin series.: A Taylor series centered at $x=0$ ; special cases include $e^x$ , $\sin x$ , $\cos x$ , and $\frac{1}{1-x}$ .
Lagrange error bound.: An upper bound on Taylor remainder: $|R_n(x)|\le \frac{\max|f^{(n+1)}(c)|}{(n+1)!}|x-a|^{n+1}$ .
Alternating series error bound.: For a convergent alternating series, the truncation error is at most the absolute value of the first omitted term.
Radius of convergence.: The value $R$ such that a power series converges for $|x-a|<R$ ; found via the Ratio Test.
Interval of convergence.: The full set of $x$ where a power series converges, including endpoint tests, expressed as an interval.
Ratio Test.: Convergence test using $L=\lim_{n\to\infty}|a_{n+1}/a_n|$ ; series converges absolutely when $L<1$ .
General term.: A closed-form expression for the $n$ th term of a series, required for full credit on FRQ series questions.

Worked example

Let

f(x)=\frac{1}{1+2x}

. (a) Write the first four nonzero terms and the general term of the Maclaurin series for

f

. (b) Find the interval of convergence. (c) Use the series to write the first four nonzero terms of the Maclaurin series for

g(x)=\int_0^x f(t)\,dt

Part (a): Start from the known geometric series

\frac{1}{1-u}=\sum_{n=0}^{\infty}u^n

. Here

\frac{1}{1+2x}=\frac{1}{1-(-2x)}

, so substitute

u=-2x

f(x)=\sum_{n=0}^{\infty}(-2x)^n=\sum_{n=0}^{\infty}(-1)^n 2^n x^n.

The first four nonzero terms are

1-2x+4x^2-8x^3

. The general term is

(-1)^n 2^n x^n

.

Part (b): A geometric series converges when

|u|<1

, i.e.

|-2x|<1

, so

|x|<\frac{1}{2}

, giving radius

R=\frac12

. Check endpoints. At

x=\frac12

: terms are

(-1)^n 2^n (1/2)^n=(-1)^n

, which does not approach 0, so it diverges. At

x=-\frac12

: terms are

(-1)^n 2^n(-1/2)^n=(-1)^n(-1)^n=1

, also failing the

n

th-term test, so it diverges. Interval of convergence:

\left(-\frac12,\frac12\right)

.

Part (c): Integrate the series term by term from

0

x

g(x)=\int_0^x \sum_{n=0}^{\infty}(-1)^n 2^n t^n\,dt=\sum_{n=0}^{\infty}(-1)^n 2^n \frac{x^{n+1}}{n+1}.

The first four nonzero terms are

x - x^2 + \frac{4}{3}x^3 - 2x^4

. (Check:

n=0

gives

x

;

n=1

gives

-2\cdot\frac{x^2}{2}=-x^2

;

n=2

gives

4\cdot\frac{x^3}{3}=\frac{4}{3}x^3

;

n=3

gives

-8\cdot\frac{x^4}{4}=-2x^4

Practice questions

The Maclaurin series for a function

f

\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n\cdot 3^n}

. What is the radius of convergence?

$R=\frac{1}{3}$
$R=1$
$R=3$
$R=\infty$

Answer: $R=3$

Apply the Ratio Test:

\lim_{n\to\infty}\left|\frac{x^{n+1}/((n+1)3^{n+1})}{x^n/(n\cdot 3^n)}\right|=\lim_{n\to\infty}\frac{n}{n+1}\cdot\frac{|x|}{3}=\frac{|x|}{3}

. Setting this less than 1 gives

|x|<3

, so the radius of convergence is

R=3

. Endpoint behavior would refine the interval but does not change the radius.

Let

P_3(x)=2-4(x-1)+3(x-1)^2-(x-1)^3

be the third-degree Taylor polynomial for

f

about

x=1

. Find

f''(1)

Answer: $f''(1)=6$

In a Taylor polynomial about

x=1

, the coefficient of

(x-1)^2

equals

\frac{f''(1)}{2!}

. Here that coefficient is

3

, so

\frac{f''(1)}{2}=3

, giving

f''(1)=6

. This tests whether you remember that Taylor coefficients embed factorials: to recover a derivative value, multiply the coefficient by

k!

The function

h

is approximated near

x=0

by its fourth-degree Maclaurin polynomial. Given that

|h^{(5)}(x)|\le 10

for all

x

[0,0.2]

, bound the error

|h(0.2)-P_4(0.2)|

Answer: $|error|\le \frac{10}{5!}(0.2)^5=\frac{10}{120}(0.00032)\approx 2.67\times10^{-5}$

Use the Lagrange error bound with

n=4

|R_4(0.2)|\le \frac{\max|h^{(5)}(c)|}{5!}|0.2-0|^5

. Substituting the given maximum

10

5!=120

, and

(0.2)^5=0.00032

gives

\frac{10\cdot 0.00032}{120}\approx 2.67\times10^{-5}

. Always write the inequality expression before the decimal to secure the justification point.

FAQ

How many points is a series FRQ usually worth?: Series free-response questions are typically worth 9 points like other FRQs, split across several parts. Because parts are graded independently and reuse a common series, you can earn substantial credit even after an early mistake — so attempt every part.
When do I use the alternating series bound versus the Lagrange error bound?: Use the alternating series error bound when the series alternates in sign with terms decreasing to zero; the error is at most the first omitted term. Use the Lagrange error bound for general Taylor approximations, especially when the series is not alternating or the problem gives you a bound on a higher derivative.
Do I always have to check both endpoints for interval of convergence?: Yes. The Ratio Test gives you the open interval, but it is inconclusive at the endpoints. You must substitute each endpoint into the series and apply a separate test (alternating series, $p$ -series, or the $n$ th-term test). Skipping endpoint checks is the most common lost point.
Can I compute derivatives directly instead of manipulating a known series?: You can, but it is slower and error-prone. For functions like $x^2 e^{-x}$ , $\sin(x^2)$ , or $\arctan x$ , start from a memorized Maclaurin series and substitute, multiply, differentiate, or integrate. This is faster and matches what graders expect for full credit.

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