AP Calc BC Series — Problem Set

85 problems spanning all 8 chapters. Each chapter has 10 problems with increasing difficulty:

• Problems 1–4: drill (formula recall)
• Problems 5–7: variations (recognize the pattern)
• Problems 8–10: AP-level (synthesis + traps)

Solutions in answers.md. Don't peek before you finish. Suggested time: 30 minutes per chapter.

Chapter 1 — Sequences

1.1 Compute $\displaystyle\lim_{n\to\infty} \frac{5n^2 - 3n + 1}{2n^2 + 7}$.

1.2 Compute $\displaystyle\lim_{n\to\infty} \frac{\ln n}{n^2}$.

1.3 Compute $\displaystyle\lim_{n\to\infty} \left(1 + \frac{2}{n}\right)^n$.

1.4 Decide whether $a_n = \frac{(-1)^n}{n+1}$ converges; if so find the limit.

1.5 Use the squeeze theorem to compute $\displaystyle\lim_{n\to\infty} \frac{\cos(n^2)}{\sqrt{n}}$.

1.6 Decide whether $a_n = \frac{n!}{3^n}$ converges.

1.7 Let $a_n = \frac{2^n}{n!}$. Show that $\{a_n\}$ is eventually strictly decreasing.

1.8 Let $a_1 = 2$ and $a_{n+1} = \sqrt{6 + a_n}$. Show the sequence converges and find its limit.

1.9 Let $a_n = \frac{1}{n} + \frac{1}{n+1} + \cdots + \frac{1}{2n}$. Find $\lim_{n\to\infty} a_n$. (Hint: relate it to an integral.)

1.10 Decide whether $a_n = \cos(n\pi)$ converges. Justify.

Chapter 2 — Series Basics, Geometric, p-Series

2.1 Find $\displaystyle\sum_{n=0}^{\infty} \frac{4}{3^n}$.

2.2 Find $\displaystyle\sum_{n=2}^{\infty} \left(\frac{1}{2}\right)^n$.

2.3 Decide whether $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{0.99}}$ converges.

2.4 Decide whether $\displaystyle\sum_{n=1}^{\infty} \frac{3n^2}{n^2 + 5}$ converges.

2.5 Find $\displaystyle\sum_{n=1}^{\infty} \frac{2^{n+1}}{5^{n-1}}$.

2.6 Find $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n(n+1)}$.

2.7 Find $\displaystyle\sum_{n=1}^{\infty} \frac{1}{(2n-1)(2n+1)}$. (Hint: partial fractions.)

2.8 Express the repeating decimal $0.\overline{37} = 0.373737\ldots$ as a fraction using a geometric series.

2.9 Given $\sum_{n=1}^{\infty} a_n = 5$ and $\sum_{n=1}^{\infty} b_n = -2$, find $\sum (3a_n + 2b_n)$.

2.10 A ball is dropped from 10 m and bounces back to $\frac{2}{3}$ of its prior height each time. Find the total distance traveled before it stops.

Chapter 3 — Convergence Tests

3.1 Use the ratio test to decide $\displaystyle\sum_{n=1}^{\infty} \frac{n^2}{2^n}$.

3.2 Use the root test to decide $\displaystyle\sum_{n=1}^{\infty} \left(\frac{3n+1}{4n+2}\right)^n$.

3.3 Use the integral test to decide $\displaystyle\sum_{n=2}^{\infty} \frac{1}{n \ln n}$.

3.4 Use the comparison test to decide $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n + n}$.

3.5 Use the limit comparison test to decide $\displaystyle\sum_{n=2}^{\infty} \frac{n+\sin n}{n^3 - 1}$.

3.6 Decide $\displaystyle\sum_{n=1}^{\infty} \frac{n!}{(2n)!}$.

3.7 Decide $\displaystyle\sum_{n=1}^{\infty} \frac{\ln n}{n^2}$.

3.8 Decide $\displaystyle\sum_{n=1}^{\infty} \frac{n^n}{n!}$. (Note the ratio limit is $e$.)

3.9 Decide $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^2 \arctan n}$.

3.10 Decide $\displaystyle\sum_{n=1}^{\infty} \frac{(2n)!}{4^n (n!)^2}$.

Chapter 4 — Alternating Series, Absolute & Conditional Convergence

4.1 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^2}$ and classify (absolute / conditional).

4.2 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n}{2n+1}$ and classify.

4.3 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n n}{n^2 + 4}$.

4.4 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} (-1)^n \cdot \frac{n}{n+1}$.

4.5 Use the AST error bound to bound the error of the 3-term partial sum of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^4}$.

4.6 How many terms of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n!}$ are needed so the partial-sum error is below $0.0001$?

4.7 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n \ln n}{n}$ and classify.

4.8 Decide convergence of $\displaystyle\sum_{n=2}^{\infty} \frac{(-1)^n}{n \ln n}$ and classify.

4.9 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n^2 + 1}}$ and classify.

4.10 Use the AST error bound to estimate $\ln 2 = \sum \frac{(-1)^{n-1}}{n}$ with error below $0.01$. How many terms?

Chapter 5 — Power Series, Radius & Interval

5.1 Find the radius and interval of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n+1}$.

5.2 Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(x-2)^n}{n^2}$.

5.3 Find the radius of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{(2x)^n}{n!}$.

5.4 Find the radius of convergence of $\displaystyle\sum_{n=1}^{\infty} n! (x+1)^n$.

5.5 Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}$.

5.6 Find the interval of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n (x-3)^n}{4^n}$.

5.7 A power series $f(x) = \sum_{n=0}^{\infty} c_n x^n$ converges at $x = 4$ and diverges at $x = -5$. For each of $x = 2, -3, 5, -7$, decide whether convergence is guaranteed, divergence is guaranteed, or undetermined.

5.8 Find the interval of convergence of $\displaystyle\sum_{n=0}^{\infty} \frac{(x+2)^n}{(n+1) \cdot 3^n}$.

5.9 Find the interval of convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n x^{2n}}{n}$. (Note: only even powers appear.)

5.10 A power series $f(x) = \sum_{n=0}^{\infty} c_n x^n$ has radius of convergence $3$. Find the radius of convergence of $g(x) = \sum c_n x^{2n}$.

Chapter 6 — Taylor / Maclaurin

6.1 Find the Maclaurin series of $f(x) = e^{2x}$.

6.2 Find the Maclaurin series of $f(x) = \sin(3x)$.

6.3 Write the degree-4 Maclaurin polynomial of $\cos x$.

6.4 Find the Taylor series of $f(x) = \ln x$ at $a = 1$.

6.5 Find the Taylor series of $f(x) = \frac{1}{x}$ at $a = 2$.

6.6 Find the Taylor series of $f(x) = e^x$ at $a = 1$.

6.7 Use the degree-4 Maclaurin polynomial to estimate $\cos(0.2)$ and compare with a calculator.

6.8 Given $f(x) = \sum c_n (x-2)^n$ with $c_3 = 5$, find $f^{(3)}(2)$.

6.9 Find the first three nonzero terms of the Taylor series of $\sin x$ at $a = \pi$.

6.10 Let $f(x) = (1+x)^{1/2}$. Write the first 4 terms of its Maclaurin series (fractional coefficients allowed).

Chapter 7 — Common Maclaurin Expansions & Operations

7.1 Find the Maclaurin series of $f(x) = e^{-x}$.

7.2 Find the Maclaurin series of $f(x) = \cos(x^2)$.

7.3 Find the Maclaurin series of $f(x) = \frac{x}{1-x}$.

7.4 Find the Maclaurin series of $f(x) = \frac{1}{(1-x)^2}$. (Hint: differentiate $\frac{1}{1-x}$.)

7.5 Use a Maclaurin series to compute $\displaystyle\lim_{x\to 0} \frac{1 - \cos x}{x^2}$.

7.6 Use a Maclaurin series to compute $\displaystyle\lim_{x\to 0} \frac{e^x - 1 - x}{x^2}$.

7.7 Find a Maclaurin-series expression for $\int_0^1 \sin(x^2)\,dx$ and estimate it using the first 2 terms.

7.8 Find an exact value for $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}$.

7.9 Find an exact value for $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n \cdot 2^n}$. (Hint: use $\ln(1+x)$ or $-\ln(1-x)$.)

7.10 Estimate $\arctan(0.1)$ using the first 3 nonzero terms of its Maclaurin series and bound the error.

Chapter 8 — Lagrange Error Bound

8.1 Find the Lagrange error bound when estimating $\sin(0.4)$ with the degree-3 Maclaurin polynomial of $\sin x$.

8.2 Find the Lagrange error bound when estimating $e^{0.3}$ with the degree-4 Maclaurin polynomial of $e^x$.

8.3 Find the Lagrange error bound when estimating $\cos(0.5)$ with the degree-6 Maclaurin polynomial of $\cos x$.

8.4 What's the smallest degree $n$ that ensures the Maclaurin polynomial of $e^x$ estimates $e$ with error below $10^{-4}$? (Use $M = 3$.)

8.5 What's the smallest degree $n$ that ensures the Maclaurin polynomial of $\sin x$ estimates $\sin(1)$ with error below $0.001$?

8.6 Show that $\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ for every real $x$ by showing the Lagrange remainder $\to 0$.

8.7 Suppose $\lvert f^{(n)}\rvert \le 5$ on $[0, 2]$ for every $n$. Bound the error of estimating $f(1)$ with the degree-5 Maclaurin polynomial.

8.8 What's the smallest degree $n$ for which the Maclaurin polynomial of $\ln(1+x)$ estimates $\ln(1.2)$ with error below $0.001$? (Hint: $f^{(n+1)}$ is messier here.)

8.9 Use the first 4 nonzero terms (i.e. $P_6$) of the Maclaurin series of $\cos x$ to estimate $\cos(\pi/4)$. Compute the estimate and the Lagrange error bound.

8.10 AP-style: let $f(x) = e^{2x}$.
(a) Write the degree-3 Maclaurin polynomial $P_3(x)$.
(b) Estimate $f(0.1)$ using $P_3$.
(c) Bound $\lvert f(0.1) - P_3(0.1) \rvert$ via Lagrange.

Cross-Chapter Challenge Problems

X.1 Decide convergence of $\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^n n}{e^n}$ and classify.

X.2 Find an exact value for $\displaystyle\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{4^{2n+1}(2n+1)!}$. (Hint: $\sin$ at a specific point.)

X.3 Find the Maclaurin series and interval of convergence of $\displaystyle f(x) = \frac{x}{1+x^2}$.

X.4 Let $f(x) = \int_0^x \frac{\sin t}{t} dt$ (with $f(0) = 0$). Find the first three nonzero terms of its Maclaurin series and its radius of convergence.

X.5 Solve the ODE $y' = y$ with $y(0) = 1$ via power series. Set $y = \sum c_n x^n$, derive a recurrence on $c_n$, and identify the closed form.

Done. Solve each chapter, then check answers.md. Suggested mastery target for AP 5: ≥ $\frac{70}{80}$ on chapters and ≥ $\frac{3}{5}$ on the challenge.