Infinite Series — AP Calculus BC Calc BC Study Guide

For: AP Calculus BC candidates sitting AP Calculus BC.

Covers: Convergence tests for infinite series, power series intervals of convergence, Taylor and Maclaurin series expansions, Lagrange error bounds, and standard series for common elementary functions.

You should already know: Strong precalculus and AB-level calculus comfort.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus BC style for educational use. They are not reproductions of past College Board papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Is Infinite Series?

An infinite series is the sum of infinitely many terms of a sequence, written formally as ${\sum }_{n=1}^{\infty }{a}_n=a_1+a_2+a_3+...$. The core question for any infinite series is whether it converges: if the partial sum $S_N={\sum }_{n=1}^N{a}_n$ approaches a finite limit as $N\to \infty$, the series converges to that limit; otherwise, it diverges. Sometimes called infinite sums or sequence sums, infinite series make up 10-15% of the AP Calculus BC exam, and are used for function approximation, numerical integration, and solving non-elementary differential equations.

2. Convergence tests — geometric, p-series, comparison

All tests below apply to series with non-negative terms, unless stated otherwise. Examiners almost always require you to name the test you use and verify its conditions to earn full marks.

Geometric Series Test

A geometric series has the form ${\sum }_{n=0}^{\infty }a{r}^n$, where $a\ne 0$ is the initial term and $r$ is the common ratio between consecutive terms. It converges if and only if $|r|<1$, with sum equal to $\frac{a}{1-r}$. If $|r|\ge 1$, the series diverges. Worked example: ${\sum }_{n=0}^{\infty }3{\left(\frac{1}{2}\right)}^n$ has $a=3$, $r=1/2$, so it converges to $\frac{3}{1-1/2}=6$.

p-Series Test

A p-series has the form ${\sum }_{n=1}^{\infty }\frac{1}{n^p}$, where $p$ is a constant. It converges if and only if $p>1$, and diverges if $p\le 1$. The harmonic series $\sum \frac{1}{n}$ (with $p=1$) is the most common divergent p-series used for comparison. Worked example: ${\sum }_{n=1}^{\infty }\frac{1}{\sqrt{n}}$ has $p=1/2<1$, so it diverges.

Comparison Tests

1. Direct Comparison Test: If $0\le a_n\le b_n$ for all $n\ge N$ (for some fixed $N$), then:

• If $\sum b_n$ converges, $\sum a_n$ converges
• If $\sum a_n$ diverges, $\sum b_n$ diverges

2. Limit Comparison Test: If ${\lim }_{n\to \infty }\frac{a_n}{b_n}=L$, where $0<L<\infty$, then $\sum a_n$ and $\sum b_n$ either both converge or both diverge. Worked example: For ${\sum }_{n=1}^{\infty }\frac{2}{n^2+4n}$, compare to $\sum \frac{1}{n^2}$ (convergent p-series, $p=2>1$). The limit of the ratio is ${\lim }_{n\to \infty }\frac{2/(n^2+4n)}{1/n^2}=2$, so the original series converges.

3. Power series and intervals of convergence

A power series is an infinite series of the form ${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$, where $a$ is the center of the series and $c_n$ are constant coefficients. Every power series falls into one of three convergence categories:

1. Converges only at the center $x=a$, with radius of convergence $R=0$
2. Converges for all real $x$, with radius of convergence $R=\infty$
3. Converges for all $x$ with $|x-a|<R$, and diverges for $|x-a|>R$, where $R>0$ is the radius of convergence.

To find $R$, use the Ratio Test: $$\underset{n\to \infty }{\lim }\left|\frac{c_{n+1}(x-a{)}^{n+1}}{c_n(x-a{)}^n}\right|=|x-a|\cdot \underset{n\to \infty }{\lim }\left|\frac{c_{n+1}}{c_n}\right|=L$$ Set $L<1$ and solve for $|x-a|<R$, where $R=\frac{1}{{\lim }_{n\to \infty }|c_{n+1}/c_n|}$ if the limit exists. You must test the endpoints $x=a+R$ and $x=a-R$ separately using standard convergence tests, as the Ratio Test is inconclusive at these points.

Worked example: Find the interval of convergence of ${\sum }_{n=1}^{\infty }\frac{(x-3{)}^n}{n\cdot 2^n}$.

• Ratio Test gives $\lim |\frac{(x-3)}{2}\cdot \frac{n}{n+1}|=\frac{|x-3|}{2}<1⟹|x-3|<2$, so $R=2$
• Test endpoint $x=5$: series becomes $\sum \frac{1}{n}$, divergent harmonic series
• Test endpoint $x=1$: series becomes $\sum \frac{(-1{)}^n}{n}$, convergent alternating harmonic series
• Final interval of convergence: $[1,5)$

4. Taylor and Maclaurin series

A Taylor series is a power series representation of a function $f(x)$ centered at a point $a$, constructed using the derivatives of $f$ at $a$. The general form is: $$\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$ where $f^{(n)}(a)$ is the $n$th derivative of $f$ evaluated at $x=a$, and $f^{(0)}(a)=f(a)$. A Maclaurin series is a special case of a Taylor series centered at $a=0$, so its form simplifies to: $$\sum _{n=0}^{\infty }\frac{f^{(n)}(0)}{n!}{x}^n$$ The $n$th-degree Taylor polynomial $P_n(x)$ is the partial sum of the first $n+1$ terms of the Taylor series, and is used to approximate $f(x)$ near $x=a$.

Worked example: Find the first 3 non-zero terms of the Maclaurin series for $f(x)=\cos (2x)$.

• Compute derivatives at $x=0$: $f(0)=1$, $f^{\prime }(0)=0$, $f^{\prime \prime }(0)=-4$, $f^{\prime \prime \prime }(0)=0$, $f^{(4)}(0)=16$
• Substitute into Maclaurin formula: $1+\frac{-4}{2!}{x}^2+\frac{16}{4!}{x}^4+...=1-2x^2+\frac{2}{3}{x}^4+...$

5. Lagrange error bound

The Lagrange error bound gives a guaranteed upper limit on the error of a Taylor polynomial approximation. If you approximate $f(x)$ with its $n$th-degree Taylor polynomial $P_n(x)$ centered at $a$, the remainder (error) $R_n(x)=f(x)-P_n(x)$ satisfies: $$|R_n(x)|\le \frac{M}{(n+1)!}|x-a{|}^{n+1}$$ where $M$ is the maximum absolute value of the $(n+1)$th derivative of $f(t)$ for all $t$ between $a$ and $x$. Examiners accept any valid upper bound for $M$, even if it is a conservative overestimate.

Worked example: Approximate $\sin (0.1)$ with the 2nd-degree Maclaurin polynomial for $\sin x$, and find the maximum error.

• 2nd-degree Maclaurin polynomial for $\sin x$ is $P_2(x)=x$ (the $x^2$ term has coefficient 0)
• The 3rd derivative of $\sin t$ is $-\cos t$, so $|f^{\prime \prime \prime }(t)|\le 1=M$ for all $t$
• $|R_2(0.1)|\le \frac{1}{3!}(0.1{)}^3=\frac{0.001}{6}\approx 0.000167$, so the approximation $\sin (0.1)\approx 0.1$ is accurate to within 0.00017.

6. Series for $e^x,\sin x,\cos x,\frac{1}{1-x}$

Memorizing these four standard Maclaurin series and their intervals of convergence will save you significant time on the exam, as you can modify them via substitution, differentiation, or integration to find series for other functions instead of computing derivatives from scratch.

1. Geometric series: $\frac{1}{1-x}={\sum }_{n=0}^{\infty }{x}^n$, valid for $|x|<1$
2. Exponential function: $e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$, valid for all real $x$
3. Sine function: $\sin x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}$, valid for all real $x$
4. Cosine function: $\cos x={\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^{2n}}{(2n)!}$, valid for all real $x$

Worked example: Find the Maclaurin series for $x{e}^{x^2}$.

• Substitute $x^2$ into the standard $e^x$ series: $e^{x^2}={\sum }_{n=0}^{\infty }\frac{(x^2{)}^n}{n!}={\sum }_{n=0}^{\infty }\frac{x^{2n}}{n!}$
• Multiply by $x$: $x{e}^{x^2}={\sum }_{n=0}^{\infty }\frac{x^{2n+1}}{n!}$, valid for all real $x$

7. Common Pitfalls (and how to avoid them)

• Wrong move: Forgetting to test endpoints of power series intervals of convergence. Why: Students stop after calculating the radius of convergence, assuming endpoints are both convergent or divergent. Correct move: Always plug in $x=a+R$ and $x=a-R$ separately, use appropriate convergence tests, and write the interval with correct brackets/parentheses for endpoints.
• Wrong move: Applying comparison tests to series with negative terms. Why: Students mix up test conditions and use direct/limit comparison for alternating series. Correct move: Only use these tests for non-negative series; for alternating series, use the Alternating Series Test or test for absolute convergence first.
• Wrong move: Using the wrong degree term in the Lagrange error bound. Why: Students confuse the degree of the Taylor polynomial with the order of the derivative for the error term. Correct move: If you use an $n$th-degree polynomial, the error bound uses the $(n+1)$th derivative, so the factorial is $(n+1)!$ and the exponent is $n+1$.
• Wrong move: Miswriting the sign pattern for $\sin x$ and $\cos x$ series. Why: Students mix up the index of the alternating $(-1{)}^n$ term. Correct move: Verify the first term to check signs: for $\sin x$, the first term is $x$, so $(-1{)}^0\frac{x^1}{1!}=x$ is correct; $(-1{)}^{n+1}$ would give $-x$, which is wrong.
• Wrong move: Assuming modified series have the same interval of convergence as the standard series. Why: Students forget substitution changes the domain of convergence for finite-radius series. Correct move: For example, substituting $2x$ into $\frac{1}{1-x}$ gives $\frac{1}{1-2x}=\sum (2x{)}^n$, which is only valid for $|2x|<1$ or $|x|<1/2$, not $|x|<1$.

8. Practice Questions (AP Calculus BC Style)

Question 1

Determine if the series ${\sum }_{n=1}^{\infty }\frac{4n^2+3}{3n^4-2n}$ converges or diverges. Justify your answer.

Solution

Use the Limit Comparison Test with the convergent p-series ${\sum }_{n=1}^{\infty }\frac{1}{n^2}$ ($p=2>1$): $$\underset{n\to \infty }{\lim }\frac{\frac{4n^2+3}{3n^4-2n}}{\frac{1}{n^2}}=\underset{n\to \infty }{\lim }\frac{4n^4+3n^2}{3n^4-2n}=\frac{4}{3}$$ The limit is positive and finite, so both series share the same convergence status. The original series converges.

Question 2

Find the interval of convergence for the power series ${\sum }_{n=0}^{\infty }\frac{(-1{)}^n{x}^n}{(n+1)\cdot 3^n}$. Show all work.

Solution

First calculate the radius of convergence using the Ratio Test: $$\underset{n\to \infty }{\lim }\left|\frac{(-1{)}^{n+1}{x}^{n+1}/[(n+2)\cdot 3^{n+1}]}{(-1{)}^n{x}^n/[(n+1)\cdot 3^n]}\right|=\underset{n\to \infty }{\lim }\left|\frac{x}{3}\cdot \frac{n+1}{n+2}\right|=\frac{|x|}{3}$$ Set $\frac{|x|}{3}<1⟹|x|<3$, so $R=3$. Test endpoints:

1. $x=3$: Series becomes ${\sum }_{n=0}^{\infty }\frac{(-1{)}^n}{n+1}$, which is the convergent alternating harmonic series.
2. $x=-3$: Series becomes ${\sum }_{n=0}^{\infty }\frac{1}{n+1}$, which is the divergent harmonic series. Final interval of convergence: $(-3,3]$.

Question 3

Use the 3rd-degree Maclaurin polynomial for $e^x$ to approximate $e^{0.3}$, and find the maximum Lagrange error bound for your approximation.

Solution

The 3rd-degree Maclaurin polynomial for $e^x$ is: $$P_3(x)=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}$$ Substitute $x=0.3$: $$P_3(0.3)=1+0.3+\frac{0.09}{2}+\frac{0.027}{6}=1+0.3+0.045+0.0045=1.3495$$ For the Lagrange error bound, the 4th derivative of $e^t$ is $e^t$. For $t$ between 0 and 0.3, $e^t\le e^{0.3}<e<3$, so $M=3$: $$|R_3(0.3)|\le \frac{3}{4!}(0.3{)}^4=\frac{3}{24}\cdot 0.0081=0.0010125$$ The approximation is accurate to within ~0.001.

9. Quick Reference Cheatsheet

10. What's Next

Mastery of infinite series directly prepares you for the remaining high-weight topics on the AP Calculus BC exam: you will use series expansions to compute integrals of non-elementary functions like $\int e^{-x^2}dx$, solve differential equations that have no closed-form algebraic solutions, and analyze parametric and polar function behavior. Series are also a foundational tool for college-level STEM courses, including real analysis, numerical methods, signal processing, and quantum physics, making this topic a critical bridge between high school and university math.

If you get stuck on any part of convergence testing, series expansion, or error bound calculation, you can ask Ollie, our AI tutor, for personalized step-by-step explanations, extra practice problems, or feedback on your work. You can also find more AP Calculus BC study materials, full-length practice exams, and topic-specific quizzes on the homepage to refine your skills ahead of test day.