AP Calculus AB/BC Study Guide & Review Unit 10 ReviewInfinite Sequences and Series (BC Only)

Unit 10 is the BC-only unit on infinite series, and its single biggest idea is that adding up infinitely many numbers can produce a finite answer. You learn a toolkit of convergence tests to decide when that happens, then use the same machinery to rewrite functions like $e^x$ and $\sin x$ as "infinite polynomials" called Taylor series. This unit is 17-18% of the exam, the largest weight of any BC unit, and it almost always anchors one full free-response question.

What this unit covers

What it means for an infinite sum to converge

• A series converges if its sequence of partial sums has a limit. The $n$th partial sum $S_n$ is just the sum of the first $n$ terms, and if $S_n \to S$ as $n \to \infty$, the series has sum $S$. Everything else in the unit builds on this definition.
• Geometric series are the one family you can actually sum exactly. If $|r| < 1$, then $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$. If $|r| \ge 1$, it diverges. Know this cold, since it powers Topic 10.15 later.
• The nth term test is a divergence-only check. If the terms don't go to 0, the series diverges. If they do go to 0, you know nothing yet (the harmonic series is the classic counterexample).

The convergence test toolkit

• The integral test connects a series to an improper integral. If $f$ is positive, continuous, and decreasing, $\sum a_n$ and $\int f(x)\,dx$ converge or diverge together. This is also where p-series come from. The p-series $\sum \frac{1}{n^p}$ converges when $p > 1$ and diverges when $p \le 1$.
• The harmonic series $\sum \frac{1}{n}$ diverges even though its terms shrink to 0. The alternating harmonic series $\sum \frac{(-1)^{n+1}}{n}$ converges. Together they're your benchmark examples for almost every test.
• Comparison test and limit comparison test let you judge an ugly series by comparing it to a clean one (usually a p-series or geometric series). Limit comparison is often easier because you only need the ratio of terms to approach a positive finite number.
• The alternating series test says an alternating series converges if its terms decrease in absolute value toward 0.
• The ratio test looks at $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right|$. Less than 1 means absolute convergence, greater than 1 means divergence, exactly 1 means the test is silent. Only these six tests (nth term, integral, comparison, limit comparison, alternating series, ratio) are assessed on the BC exam.

Absolute vs. conditional convergence, and bounding error

• A series converges absolutely if $\sum |a_n|$ converges, and conditionally if it converges but $\sum |a_n|$ diverges. Absolute convergence implies convergence, and absolutely convergent series can be rearranged without changing the sum.
• The Alternating Series Error Bound says the error from stopping a convergent alternating series after $n$ terms is at most the absolute value of the first unused term. Simple statement, frequent FRQ points.
• The Lagrange Error Bound does the same job for Taylor polynomial approximations of any series. It caps the error using the maximum size of the next derivative on the interval.

Taylor series, Maclaurin series, and power series

• A Taylor polynomial rebuilds a function near $x = a$ using its derivatives. The coefficient of the degree-$n$ term is $\frac{f^{(n)}(a)}{n!}$. As the degree grows, the polynomial typically hugs the function over a wider interval.
• A Taylor series is the infinite version, and a Taylor polynomial is just one of its partial sums. A Maclaurin series is a Taylor series centered at 0.
• Memorize the big three Maclaurin series for $e^x$, $\sin x$, and $\cos x$, plus the geometric series for $\frac{1}{1-x}$. Nearly every series you're asked to build comes from manipulating these.
• Power series have the form $\sum a_n(x - r)^n$. Use the ratio test to find the radius of convergence $R$, then test both endpoints separately to nail down the interval of convergence. A power series converges at a single point, on an interval, or everywhere.
• You can build new power series from known ones by substitution (swap $x$ for $-x^2$, say), multiplication, term-by-term differentiation, and term-by-term integration. This is how you get series for things like $\arctan x$ or $\ln(1+x)$ from the geometric series.

Infinite Sequences and Series (BC only) at a glance

Why Infinite Sequences and Series (BC only) matters in AP Calc

Series is where the Limits big idea gets its biggest payoff. Convergence is literally defined as a limit of partial sums, so the unit takes the very first concept of the course and pushes it to its most powerful conclusion, that polynomials (the easiest functions in math) can stand in for transcendental functions like $e^x$ with controllable error.

• It's the heaviest-weighted unit on the BC exam at 17-18%, and historically one of the six free-response questions is devoted to series.
• It answers a question your calculator quietly raises all year, which is how a machine actually computes $\sin(0.3)$ or $e^{1.2}$. The answer is Taylor polynomials plus an error bound.
• It rewards justification skills. Naming the right test, verifying its conditions, and writing the conclusion correctly is exactly the kind of reasoning the exam scores point by point.

How this unit connects across the course

• Limits and Continuity (Unit 1) supplies the foundation. Convergence of a series is the limit of its partial sums, and every test ends with evaluating a limit.
• Differentiation (Units 2-3) feeds Taylor polynomials directly, since every coefficient $\frac{f^{(n)}(a)}{n!}$ comes from a higher-order derivative. Taylor polynomials are also the natural upgrade of tangent line approximation from Unit 4.
• Integration (Unit 6) powers the integral test, which uses improper integrals to settle convergence, and term-by-term integration is how you generate new power series from known ones.
• Differential Equations (Unit 7) and Taylor series meet on FRQs. A series question can hand you a differential equation and ask you to build the Taylor series of its solution, and Euler's method error ideas echo the error-bound mindset here.

Key formulas and procedures

• Geometric series sum, $\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}$ when $|r| < 1$. The only formula that hands you an exact value of an infinite sum.
• p-series rule, $\sum \frac{1}{n^p}$ converges iff $p > 1$. Your go-to comparison benchmark.
• nth term test, if $\lim_{n\to\infty} a_n \neq 0$ the series diverges. A 10-second first check on any series.
• Ratio test, compute $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$; $L < 1$ converges absolutely, $L > 1$ diverges. The default tool for factorials and for radius of convergence.
• Taylor polynomial centered at $a$, $P_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k$. Approximates $f$ near $x = a$.
• Maclaurin series to memorize, $e^x = \sum \frac{x^n}{n!}$, $\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $\cos x = \sum \frac{(-1)^n x^{2n}}{(2n)!}$, $\frac{1}{1-x} = \sum x^n$ for $|x|<1$. Building blocks for every other series.
• Alternating Series Error Bound, $|S - S_n| \le |a_{n+1}|$. The error is no bigger than the first term you left off.
• Lagrange Error Bound, $|R_n(x)| \le \frac{\max|f^{(n+1)}|}{(n+1)!}|x-a|^{n+1}$. Caps Taylor polynomial error when the series doesn't alternate nicely.
• Interval of convergence procedure. Apply the ratio test, set $L < 1$, solve for $x$ to get the radius, then plug each endpoint back in and test it with another method. Endpoint checks are separate points on FRQs.
• Term-by-term differentiation and integration of power series. Differentiate or integrate inside the sum to build new series; the radius of convergence stays the same.

Infinite Sequences and Series (BC only) on the AP exam

This unit is 17-18% of the BC exam, the largest single share of any unit, and it's tested in both multiple choice and free response. One free-response question is typically a dedicated series question, and it usually layers several skills in one problem.

• A classic FRQ arc looks like this. Build a Taylor polynomial from given derivative values or a defining rule, use it to approximate a function value, then bound the error with the Alternating Series Error Bound or Lagrange Error Bound and justify the bound in words.
• Convergence questions demand named justification. Saying "it converges by the ratio test because the limit is 1/3, which is less than 1" earns the point; an unexplained answer doesn't. Verify the test's conditions before using its conclusion.
• Radius and interval of convergence shows up regularly. Expect to run the ratio test on a power series and to check both endpoints separately, since the open interval alone is not a complete answer.
• Multiple choice loves quick test selection. You'll see a series and need to recognize in seconds whether nth term, p-series, geometric, comparison, alternating, or ratio is the fastest path, plus classify series as absolutely convergent, conditionally convergent, or divergent.

Essential questions

• How can adding infinitely many numbers produce a finite total, and what exactly does "the sum" mean?
• Why do some series with terms shrinking to zero still diverge, and how do the tests tell the difference?
• What does it mean to say a polynomial "becomes" a function like $e^x$, and on what interval is that claim true?
• How do you guarantee an approximation is good enough when you can never add all the terms?

Key terms to know

• Partial sum: The sum $S_n$ of the first $n$ terms of a series, whose limit defines convergence.
• Convergent series: A series whose partial sums approach a finite limit $S$, called the sum of the series.
• Geometric series: A series with a constant ratio $r$ between consecutive terms, converging to $\frac{a}{1-r}$ when $|r| < 1$.
• Harmonic series: The series $\sum \frac{1}{n}$, which diverges even though its terms go to 0.
• p-series: A series of the form $\sum \frac{1}{n^p}$, convergent exactly when $p > 1$.
• Absolute convergence: When $\sum |a_n|$ converges, which guarantees the original series converges and can be safely rearranged.
• Conditional convergence: When a series converges but the series of absolute values diverges, like the alternating harmonic series.
• Power series: A series of the form $\sum a_n(x-r)^n$, a polynomial with infinitely many terms centered at $x = r$.
• Radius of convergence: The distance $R$ from the center within which a power series is guaranteed to converge, found with the ratio test.
• Interval of convergence: The full set of $x$ values where a power series converges, found by adding endpoint checks to the radius.
• Taylor polynomial: A degree-$n$ polynomial matching $f$ and its first $n$ derivatives at $x = a$, used to approximate $f$ nearby.
• Maclaurin series: A Taylor series centered at $x = 0$, including the standard series for $e^x$, $\sin x$, $\cos x$, and $\frac{1}{1-x}$.
• Alternating Series Error Bound: The guarantee that truncating a convergent alternating series leaves an error no larger than the first omitted term.
• Lagrange Error Bound: An upper bound on Taylor polynomial error built from the maximum size of the next derivative.

Common mix-ups

• The nth term test never proves convergence. Terms going to 0 is necessary but not sufficient, and the harmonic series is the proof. Writing "$a_n \to 0$, so it converges" loses points every time.
• A ratio test result of $L = 1$ is inconclusive, not divergent. You have to switch to a different test, which is exactly why endpoint checks for power series need a separate method.
• The radius of convergence gives you an open interval, but the interval of convergence may include one endpoint, both, or neither. Always test each endpoint individually.
• A Taylor polynomial and a Taylor series are not the same thing. The polynomial is a finite partial sum of the series, which is why error bounds exist in the first place.