AP Calculus BC Unit 10: Infinite Sequences and Series

When does adding infinitely many numbers give a finite total? This unit builds a toolkit of convergence tests (each with conditions you must state), error bounds for approximations, and power series: Taylor and Maclaurin series, their intervals of convergence, and new series made from known ones.

BC topics on this page: 10.1 Convergent and divergent series; 10.2 Geometric series; 10.3 The nth-term test; 10.4 The integral test; 10.5 Harmonic series and p-series; 10.6 Comparison tests; 10.7 Alternating series test; 10.8 Ratio test; 10.9 Absolute and conditional convergence; 10.10 Alternating series error bound; 10.11 Taylor polynomials; 10.12 Lagrange error bound; 10.13 Radius and interval of convergence; 10.14 Taylor and Maclaurin series; 10.15 Power series of new functions.

Key formulas and facts

• $\sum a_n$ converges to $S$ when the partial sums $S_n$ tend to $S$; geometric: $\sum_{n=0}^\infty ar^n=\frac{a}{1-r}$ for $|r|<1$, divergent for $|r|\ge1$
• $n$th-term test: if $a_n\not\to0$, the series diverges (if $a_n\to0$, no conclusion)
• $p$-series $\sum\frac{1}{n^p}$: converges for $p>1$, diverges for $p\le1$; integral test: for positive, continuous, decreasing $f$ with $f(n)=a_n$, $\sum a_n$ and $\int_1^\infty f(x)\,dx$ both converge or both diverge
• Comparison: $0\le a_n\le b_n$ and $\sum b_n$ converges gives convergence; limit comparison: $\lim\frac{a_n}{b_n}=L$ with $0<L<\infty$ means both behave the same
• Alternating series test: $b_n>0$ decreasing with $b_n\to0$ gives convergence of $\sum(-1)^nb_n$, and $|S-S_n|\le b_{n+1}$
• Ratio test: $L=\lim\left|\frac{a_{n+1}}{a_n}\right|$; $L<1$ converges absolutely, $L>1$ diverges, $L=1$ inconclusive
• Taylor polynomial: $P_n(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k$; Lagrange bound: $|f(x)-P_n(x)|\le\frac{\max|f^{(n+1)}|}{(n+1)!}|x-a|^{n+1}$
• Known series: $e^x=\sum\frac{x^n}{n!}$, $\sin x=\sum\frac{(-1)^nx^{2n+1}}{(2n+1)!}$, $\cos x=\sum\frac{(-1)^nx^{2n}}{(2n)!}$ (all $x$); $\frac1{1-x}=\sum x^n$ ($|x|<1$)
• Interval of convergence: find the radius with the ratio test, then test each endpoint separately with another test

Worked examples

• Telescoping: $\sum_{n=1}^\infty\left(\frac1n-\frac1{n+2}\right)=1+\frac12=\frac32$
• Geometric: $\sum_{n=1}^\infty\frac{2}{5^n}=\frac{2/5}{1-1/5}=\frac12$
• Ratio test: $\sum\frac{n^2}{2^n}$ has $\frac{(n+1)^2}{2n^2}\to\frac12<1$, so it converges
• Conditional: $\sum\frac{(-1)^n}{\sqrt n}$ converges by the alternating series test, but $\sum\frac1{\sqrt n}$ diverges
• Taylor: for $f(x)=\ln(1+x)$, $P_3(x)=x-\frac{x^2}{2}+\frac{x^3}{3}$
• Lagrange: $e^{0.1}\approx1+0.1=1.1$ with $|f''|\le e^{0.1}<2$ gives error at most $\frac{2}{2!}(0.1)^2=0.01$
• Interval: $\sum\frac{x^n}{n\,3^n}$ has radius $3$; at $x=3$ it is harmonic (diverges), at $x=-3$ alternating harmonic (converges): $-3\le x<3$
• New series: $x^2\cos x=x^2-\frac{x^4}{2}+\frac{x^6}{24}-\cdots$

Common traps

• Using the $n$th-term test to prove convergence: $a_n\to0$ never shows a series converges.
• Forgetting to test the endpoints of an interval of convergence, or testing them with the ratio test (it always gives $1$ there).
• Stating 'by the alternating series test' without checking that the terms decrease and tend to $0$.
• Using the alternating series error bound for a series that is not alternating, or the wrong term: the bound is the first omitted term.
• Writing $\frac{f^{(n)}(a)}{n}$ instead of $\frac{f^{(n)}(a)}{n!}$ in a Taylor coefficient.
• In a Lagrange bound, using $f^{(n)}$ instead of $f^{(n+1)}$, or not justifying the maximum $M$ on the interval.
• Comparing in the wrong direction: being smaller than a divergent series, or larger than a convergent one, proves nothing.

Practice questions

105 multiple-choice questions and 3 free-response questions. Work each one out before you open the answer. In a free-response scoring guide, each line is worth the points it states. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The $n$th partial sum of a series $\sum a_n$ is $S_n=\dfrac{2n}{n+3}$. What is the sum of the series?

1. $\frac{2}{3}$
2. $\frac{1}{2}$
3. $2$
4. The series diverges.

Question 2 · Multiple choice

What is the sum of the telescoping series $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+1)}$?

1. $\frac{1}{2}$
2. $1$
3. $2$
4. The series diverges.

Question 3 · Multiple choice

What is the value of $\displaystyle\sum_{n=1}^{\infty}\frac{2}{n(n+2)}$?

1. $1$
2. $2$
3. $\frac{1}{2}$
4. $\frac{3}{2}$

Question 4 · Multiple choice

The partial sums of $\sum a_n$ are $S_n=\dfrac{n^2}{n+1}$. What is $a_3$?

1. $\frac{9}{4}$
2. $\frac{4}{3}$
3. $\frac{11}{12}$
4. $\frac{3}{4}$

Question 5 · Multiple choice

What is the value of $\displaystyle\sum_{n=1}^{\infty}\left(\frac1n-\frac{1}{n+3}\right)$?

1. $\frac{11}{6}$
2. $1$
3. $\frac{3}{2}$
4. The series diverges.

Question 6 · Multiple choice

What is $\displaystyle\sum_{n=1}^{\infty}\ln\left(\frac{n}{n+1}\right)$?

1. $0$
2. $- \ln{\left(2 \right)}$
3. $-1$
4. The series diverges.

Question 7 · Multiple choice

The partial sums of $\sum a_n$ are $S_n=4-\dfrac{3}{2^n}$. What is $\displaystyle\sum_{n=3}^{\infty}a_n$?

1. $4$
2. $\frac{3}{4}$
3. $\frac{3}{8}$
4. $\frac{13}{4}$

Question 8 · Multiple choice

What is $\displaystyle\sum_{n=0}^{\infty}5\left(\frac23\right)^n$?

1. $10$
2. $\frac{5}{3}$
3. The series diverges.
4. $15$

Question 9 · Multiple choice

What is $\displaystyle\sum_{n=1}^{\infty}3\left(-\frac12\right)^n$?

1. $2$
2. $-1$
3. $- \frac{3}{2}$
4. $1$

Question 10 · Multiple choice

What is $\displaystyle\sum_{n=2}^{\infty}\frac{4}{3^n}$?

1. $\frac{2}{3}$
2. $2$
3. $\frac{4}{9}$
4. $\frac{4}{3}$

Question 11 · Multiple choice

Which of the following series converge? I. $\displaystyle\sum_{n=1}^{\infty}\left(\frac54\right)^n$ II. $\displaystyle\sum_{n=1}^{\infty}(-0.9)^n$ III. $\displaystyle\sum_{n=1}^{\infty}\frac{2^n}{3^{n+1}}$

1. III only
2. I and III only
3. II and III only
4. I, II and III

Question 12 · Multiple choice

For which values of $x$ does the geometric series $\displaystyle\sum_{n=0}^{\infty}\frac{(x-1)^n}{2^n}$ converge?

1. $-1\le x\le3$
2. $-1<x<3$
3. $0<x<2$
4. $-3<x<1$

Question 13 · Multiple choice

The repeating decimal $0.272727\ldots$ equals $\displaystyle\sum_{n=1}^{\infty}\frac{27}{100^n}$. What fraction is it?

1. $\frac{27}{100}$
2. $\frac{27}{101}$
3. $\frac{2}{7}$
4. $\frac{3}{11}$

Question 14 · Multiple choice

A ball is dropped from a height of $12$ feet. Each time it hits the floor it rebounds to $\frac34$ of the height of its previous fall. What is the total vertical distance the ball travels?

1. $48$
2. $96$
3. $84$
4. $72$

Question 15 · Multiple choice

Which of the following series diverges by the $n$th-term test?

1. $\displaystyle\sum_{n=1}^{\infty}\frac{n}{2n+1}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}$
3. $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$
4. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{2^n}$

Question 16 · Multiple choice

Suppose $\displaystyle\lim_{n\to\infty}a_n=0$. Which statement must be true?

1. $\sum a_n$ may converge or may diverge.
2. $\sum a_n$ converges.
3. $\sum a_n$ diverges.
4. $\sum a_n$ converges to $0$.

Question 17 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}\frac{n^2+1}{3n^2-n}$?

1. The series converges, because $\lim_{n\to\infty}a_n=\frac13$ is finite.
2. The series converges by the alternating series test.
3. The series diverges, because $\lim_{n\to\infty}a_n=\frac13\ne0$.
4. The nth-term test is inconclusive, so no conclusion can be drawn.

Question 18 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}\cos\left(\frac1n\right)$?

1. The series converges, because $\lim_{n\to\infty}a_n=1$ is finite.
2. The series diverges, because $\lim_{n\to\infty}a_n=1\ne0$.
3. The series converges by the alternating series test.
4. The nth-term test is inconclusive, so no conclusion can be drawn.

Question 19 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}\left(1+\frac1n\right)^n$?

1. The series converges, because $\lim_{n\to\infty}a_n=e$ is finite.
2. The series converges by the alternating series test.
3. The nth-term test is inconclusive, so no conclusion can be drawn.
4. The series diverges, because $\lim_{n\to\infty}a_n=e\ne0$.

Question 20 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}n\sin\left(\frac1n\right)$?

1. The series converges, because $\lim_{n\to\infty}a_n=1$ is finite.
2. The series converges by the alternating series test.
3. The series diverges, because $\lim_{n\to\infty}a_n=1\ne0$.
4. The nth-term test is inconclusive, so no conclusion can be drawn.

Question 21 · Multiple choice

Why does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n\,n}{n+1}$ diverge?

1. Its terms do not approach $0$: $\left|a_n\right|\to1$.
2. Its terms do not alternate in sign.
3. It is a $p$-series with $p\le1$.
4. The ratio test gives a limit greater than $1$.

Question 22 · Multiple choice

The integral test is applied to $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n\ln n}$. What is the result?

1. The series converges, because $\frac{1}{x\ln x}\to0$.
2. The series converges, because $\int_2^\infty\frac{dx}{x\ln x}=\frac{1}{\ln2}$.
3. The series diverges, because $\frac{1}{x\ln x}$ is not decreasing.
4. The series diverges, because $\int_2^\infty\frac{dx}{x\ln x}=\lim_{b\to\infty}\left(\ln(\ln b)-\ln(\ln2)\right)=\infty$.

Question 23 · Multiple choice

To apply the integral test to $\displaystyle\sum_{n=1}^{\infty}ne^{-n^2}$, a student evaluates $\displaystyle\int_1^\infty xe^{-x^2}\,dx$. What is its value?

1. $e^{-1}$
2. $\frac{1}{2 e}$
3. $\frac{1}{2}$
4. The series diverges.

Question 24 · Multiple choice

What is the value of $\displaystyle\int_1^\infty\frac{dx}{x^2+1}$, which the integral test uses for $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2+1}$?

1. $\frac{\pi}{2}$
2. $1$
3. The series diverges.
4. $\frac{\pi}{4}$

Question 25 · Multiple choice

What does the integral test show about $\displaystyle\sum_{n=1}^{\infty}\frac{\ln n}{n}$?

1. It converges, because $\frac{\ln n}{n}\to0$.
2. It diverges, because $\int_3^\infty\frac{\ln x}{x}\,dx=\lim_{b\to\infty}\frac{(\ln b)^2-(\ln3)^2}{2}=\infty$.
3. It converges, because $\int_3^\infty\frac{\ln x}{x}\,dx$ is finite.
4. The integral test cannot be used, because $\frac{\ln x}{x}$ is never decreasing.

Question 26 · Multiple choice

For $S=\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^3}$, the integral test gives $S\le a_1+\displaystyle\int_1^\infty\frac{dx}{x^3}$. What is this upper bound?

1. $\frac{3}{2}$
2. $\frac{1}{2}$
3. $1$
4. $2$

Question 27 · Multiple choice

What is $\displaystyle\int_2^\infty\frac{dx}{x(\ln x)^2}$, used to test $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n(\ln n)^2}$?

1. $\ln{\left(2 \right)}$
2. $\frac{1}{\ln{\left(2 \right)}^{2}}$
3. $\frac{1}{\ln{\left(2 \right)}}$
4. The series diverges.

Question 28 · Multiple choice

For which values of $p>0$ does $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n(\ln n)^p}$ converge?

1. $p\ge1$ only
2. $p>1$ only
3. $0<p<1$ only
4. All $p>0$

Question 29 · Multiple choice

Which of the following series converges?

1. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{0.99}}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac1n$
3. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt n}$
4. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{1.1}}$

Question 30 · Multiple choice

For which values of $k$ does $\displaystyle\sum_{n=1}^{\infty}\frac{n^k}{n^3}$ converge?

1. $k<3$
2. $k\le2$
3. $k<2$
4. $k>2$

Question 31 · Multiple choice

Given that $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$, what is $\displaystyle\sum_{n=2}^{\infty}\frac{1}{n^2}$?

1. $-1 + \frac{\pi^{2}}{6}$
2. $\frac{\pi^{2}}{6}$
3. $- \frac{1}{4} + \frac{\pi^{2}}{6}$
4. $\frac{\pi^{2}}{24}$

Question 32 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^{\pi/3}}$?

1. It converges, because it is a $p$-series with $p=\frac\pi3>1$.
2. It diverges, because it is a $p$-series with $p=\frac\pi3<1$.
3. It diverges by the $n$th-term test.
4. It converges by the alternating series test.

Question 33 · Multiple choice

What can be concluded about $\displaystyle\sum_{n=1}^{\infty}\left(\frac1n-\frac1{n^2}\right)$?

1. It converges, because both $\sum\frac1n$ and $\sum\frac1{n^2}$ are $p$-series.
2. It converges, because its terms tend to $0$.
3. It diverges: it is the harmonic series (divergent) minus a convergent $p$-series.
4. It converges to $0$, because the two series cancel.

Question 34 · Multiple choice

What is the fourth partial sum $S_4$ of the harmonic series $\displaystyle\sum_{n=1}^{\infty}\frac1n$?

1. $\frac{13}{6}$
2. $\frac{25}{12}$
3. $\frac{11}{6}$
4. $4$

Question 35 · Multiple choice

For which values of $p$ does the $p$-series $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^p}$ diverge?

1. $p<1$
2. $p>1$
3. $p\ge1$
4. $p\le1$

Question 36 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2+5}$ is true?

1. It diverges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
2. It converges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
3. It converges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}$.
4. It diverges by the $n$th-term test.

Question 37 · Multiple choice

The limit comparison test is used on $\displaystyle\sum_{n=1}^{\infty}\frac{1}{2n-1}$ and $\displaystyle\sum_{n=1}^{\infty}\frac1n$. What is $\displaystyle\lim_{n\to\infty}\frac{1/(2n-1)}{1/n}$?

1. $\frac{1}{2}$
2. $2$
3. $1$
4. $0$

Question 38 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{n+1}{n^3+2}$ is true?

1. It diverges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
2. It converges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
3. It diverges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}$.
4. It converges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}$.

Question 39 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+1}}$ is true?

1. It converges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}$.
2. It diverges, by limit comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
3. It converges, because its terms tend to $0$.
4. It diverges by the $n$th-term test.

Question 40 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{2^n}{3^n+1}$ is true?

1. It diverges, by comparison with $\displaystyle\sum_{n=1}^{\infty}2^n$.
2. It diverges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
3. It converges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1n$.
4. It converges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\left(\frac23\right)^n$.

Question 41 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{\sin^2 n}{n^2}$ is true?

1. It diverges, because $\frac{\sin^2n}{n^2}\le\frac1{n^2}$.
2. It converges, because $0\le\frac{\sin^2n}{n^2}\le\frac1{n^2}$ and $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}$ converges.
3. It converges by the alternating series test.
4. It diverges, because $\sin^2n$ has no limit.

Question 42 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n-\frac12}$ is true?

1. It diverges, because $\frac{1}{n-\frac12}>\frac1n$ and $\displaystyle\sum_{n=1}^{\infty}\frac1n$ diverges.
2. It converges, because $\frac{1}{n-\frac12}>\frac1n$.
3. It converges, because $\frac{1}{n-\frac12}<\frac{1}{n^2}$.
4. It diverges by the $n$th-term test.

Question 43 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{\sqrt n}$ is true?

1. It diverges, because $\displaystyle\sum_{n=1}^{\infty}\frac{1}{\sqrt n}$ diverges.
2. It diverges by the $n$th-term test.
3. It converges by the alternating series test, because $\frac1{\sqrt n}$ decreases to $0$.
4. It converges by the ratio test.

Question 44 · Multiple choice

What is the value of $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$?

1. $1$
2. $\ln{\left(2 \right)}$
3. $\frac{1}{2}$
4. The series diverges.

Question 45 · Multiple choice

Recognize $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}$ as a known Maclaurin series evaluated at a point. To what number does it converge?

1. $\frac{\pi}{2}$
2. $1$
3. $\ln{\left(2 \right)}$
4. $\frac{\pi}{4}$

Question 46 · Multiple choice

Which statement about $\displaystyle\sum_{n=2}^{\infty}\frac{(-1)^n}{\ln n}$ is true?

1. It diverges, because $\frac{1}{\ln n}>\frac1n$.
2. It diverges by the $n$th-term test.
3. It converges by the alternating series test.
4. It converges by the ratio test.

Question 47 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n\,n}{n^2+1}$ is true?

1. It converges by the alternating series test.
2. It diverges, because $\frac{n}{n^2+1}$ behaves like $\frac1n$.
3. It diverges by the $n$th-term test.
4. It converges by the ratio test.

Question 48 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}(-1)^n e^{1/n}$ is true?

1. It diverges, because $\left|a_n\right|=e^{1/n}\to1\ne0$.
2. It converges by the alternating series test.
3. It converges, because $e^{1/n}$ is decreasing.
4. It converges by the ratio test.

Question 49 · Multiple choice

Which series meets every condition of the alternating series test: terms that alternate in sign and whose absolute values decrease to $0$?

1. $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n(n+1)}{n}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac{\sin n}{n}$
3. $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{2n+3}$
4. $\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{n}{n+5}$

Question 50 · Multiple choice

For $\displaystyle\sum_{n=1}^{\infty}\frac{n}{3^n}$, what is $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$?

1. $3$
2. $\frac{1}{3}$
3. $1$
4. $0$

Question 51 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{n!}{2^n}$ is true?

1. It converges, because $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac12$.
2. It converges, by comparison with $\displaystyle\sum_{n=1}^{\infty}\frac1{2^n}$.
3. The ratio test is inconclusive.
4. It diverges, because $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\infty$.

Question 52 · Multiple choice

For $\displaystyle\sum_{n=1}^{\infty}\frac{5^n}{n!}$, what is $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$?

1. $5$
2. $1$
3. $0$
4. $\frac{1}{5}$

Question 53 · Multiple choice

For which series is the ratio test inconclusive?

1. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^2}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{2^n}$
3. $\displaystyle\sum_{n=1}^{\infty}\frac{3^n}{n}$
4. $\displaystyle\sum_{n=1}^{\infty}\frac{n^2}{4^n}$

Question 54 · Multiple choice

For $\displaystyle\sum_{n=1}^{\infty}\frac{n^n}{n!}$, what is $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$?

1. $1$
2. $e^{-1}$
3. $\infty$
4. $e$

Question 55 · Multiple choice

For $\displaystyle\sum_{n=1}^{\infty}\frac{(2n)!}{(n!)^2\,5^n}$, what is $\displaystyle\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$?

1. $\frac{2}{5}$
2. $\frac{4}{5}$
3. $\frac{1}{5}$
4. $4$

Question 56 · Multiple choice

Which statement about $\displaystyle\sum_{n=1}^{\infty}\frac{n^3}{3^n}$ is true?

1. It diverges, because $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=3$.
2. The ratio test is inconclusive.
3. It diverges by the $n$th-term test.
4. It converges, because $\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|=\frac13$.

Question 57 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n}$ behave?

1. Absolutely convergent
2. Conditionally convergent
3. Divergent
4. Convergent, but the tests do not decide which kind

Question 58 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$ behave?

1. Absolutely convergent
2. Conditionally convergent
3. Divergent
4. Convergent, but the tests do not decide which kind

Question 59 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n\,n}{n+2}$ behave?

1. Absolutely convergent
2. Conditionally convergent
3. Divergent
4. Convergent, but the tests do not decide which kind

Question 60 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt[3]{n}}$ behave?

1. Absolutely convergent
2. Conditionally convergent
3. Divergent
4. Convergent, but the tests do not decide which kind

Question 61 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\left(-\frac23\right)^n$ behave?

1. Conditionally convergent
2. Divergent
3. Convergent, but the tests do not decide which kind
4. Absolutely convergent

Question 62 · Multiple choice

How does $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2+1}$ behave?

1. Conditionally convergent
2. Divergent
3. Absolutely convergent
4. Convergent, but the tests do not decide which kind

Question 63 · Multiple choice

Which of the following series converges conditionally?

1. $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}$
2. $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$
3. $\displaystyle\sum_{n=1}^{\infty}(-1)^n$
4. $\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}$

Question 64 · Multiple choice

The sum $S=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2}$ is approximated by the partial sum $S_4$. By the alternating series error bound, what is the smallest bound this guarantees for $|S-S_4|$?

1. $\frac{1}{25}$
2. $\frac{1}{16}$
3. $\frac{1}{5}$
4. $\frac{1}{36}$

Question 65 · Multiple choice

What is the smallest number of terms $n$ for which the alternating series error bound guarantees that $|S-S_n|<0.01$, where $S=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}$?

1. $99$
2. $101$
3. $100$
4. $10$

Question 66 · Multiple choice

The approximation $\sin(0.5)\approx0.5-\dfrac{0.5^3}{3!}$ uses the alternating Maclaurin series of $\sin x$. What bound does the alternating series error bound give for its error?

1. $\frac{1}{48}$
2. $\frac{1}{3840}$
3. $\frac{1}{384}$
4. $\frac{1}{120}$

Question 67 · Multiple choice

The series $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}$ equals $e^{-1}$. The partial sum using the terms for $n=0,1,2,3$ is $\frac13$. What error bound does the alternating series error bound give?

1. $\frac{1}{6}$
2. $\frac{1}{120}$
3. $\frac{1}{3}$
4. $\frac{1}{24}$

Question 68 · Multiple choice

Let $S=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^3}$, and let $S_3=1-\frac18+\frac1{27}$. Which statement is true?

1. $S_3<S$, because the first omitted term is negative.
2. $S_3=S$, because the series converges absolutely.
3. $S_3>S$, because the first omitted term is negative.
4. $S_3>S$, because the first omitted term is positive.

Question 69 · Multiple choice

The sum $S=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{2^n}$ is estimated by $S_2=\frac12-\frac14$. Using the alternating series error bound, which interval must contain $S$?

1. $\left[\frac14,\frac38\right]$
2. $\left[\frac18,\frac14\right]$
3. $\left[\frac14,\frac12\right]$
4. $\left[\frac{3}{16},\frac{5}{16}\right]$

Question 70 · Multiple choice

The approximation $\cos(0.2)\approx1-\dfrac{0.2^2}{2}=0.98$ comes from the Maclaurin series of $\cos x$. What is the alternating series error bound for it?

1. $\frac{1}{500}$
2. $\frac{1}{1500}$
3. $\frac{1}{150000}$
4. $\frac{1}{15000}$

Question 71 · Multiple choice

A function $f$ has $f(0)=2$, $f'(0)=-1$, $f''(0)=6$ and $f'''(0)=12$. What is the third-degree Taylor polynomial for $f$ about $x=0$?

1. $2-x+6x^2+12x^3$
2. $2-x+3x^2+2x^3$
3. $2-x+3x^2+4x^3$
4. $2+x+3x^2+2x^3$

Question 72 · Multiple choice

What is the second-degree Taylor polynomial for $\sqrt x$ about $x=4$?

1. $2+\frac{x-4}{4}-\frac{(x-4)^2}{32}$
2. $2+\frac{x-4}{2}-\frac{(x-4)^2}{8}$
3. $2+\frac{x-4}{4}+\frac{(x-4)^2}{64}$
4. $2+\frac{x-4}{4}-\frac{(x-4)^2}{64}$

Question 73 · Multiple choice

Using the third-degree Maclaurin polynomial for $e^x$, what is the approximation of $e^{0.2}$?

1. $\frac{61}{50}$
2. $\frac{458}{375}$
3. $\frac{11}{9}$
4. $\frac{6}{5}$

Question 74 · Multiple choice

What is the coefficient of $(x-1)^3$ in the Taylor polynomial for $\ln x$ about $x=1$?

1. $\frac{1}{3}$
2. $2$
3. $- \frac{1}{3}$
4. $\frac{1}{6}$

Question 75 · Multiple choice

The Taylor polynomial of degree $3$ for $f$ about $x=2$ is $P(x)=3-2(x-2)+5(x-2)^2-(x-2)^3$. What is $f''(2)$?

1. $5$
2. $\frac{5}{2}$
3. $10$
4. $-6$

Question 76 · Multiple choice

What is the second-degree Maclaurin polynomial for $\cos(3x)$?

1. $1-\frac{3x^2}{2}$
2. $1-\frac{9x^2}{2}$
3. $1-9x^2$
4. $1+\frac{9x^2}{2}$

Question 77 · Multiple choice

The table gives derivatives of $f$ at $x=1$. Using the third-degree Taylor polynomial for $f$ about $x=1$, what is the approximation of $f(1.5)$?

1. $3.375$
2. $2.625$
3. $3.0$
4. $3.125$

Question 78 · Multiple choice

The third-degree Maclaurin polynomial for $\sin x$ is used to approximate $\sin(0.3)$. Using $\left|f^{(4)}(x)\right|\le1$, what is the Lagrange error bound?

1. $\frac{9}{2000}$
2. $\frac{27}{40000}$
3. $\frac{27}{80000}$
4. $\frac{1}{2400}$

Question 79 · Multiple choice

The second-degree Maclaurin polynomial for $e^x$ is used to approximate $e^{0.5}$. Using $\left|f'''(x)\right|\le2$ for $0\le x\le0.5$, what is the Lagrange error bound?

1. $\frac{1}{24}$
2. $\frac{1}{12}$
3. $\frac{1}{48}$
4. $\frac{1}{8}$

Question 80 · Multiple choice

A function has $\left|f^{(4)}(x)\right|\le6$ for $1\le x\le1.4$. The third-degree Taylor polynomial about $x=1$ is used to approximate $f(1.4)$. What is the Lagrange error bound?

1. $\frac{4}{625}$
2. $\frac{8}{375}$
3. $\frac{16}{625}$
4. $\frac{1}{250}$

Question 81 · Multiple choice

To approximate $e$ with a Maclaurin polynomial for $e^x$ at $x=1$, use $\left|f^{(n+1)}(x)\right|\le3$ on $[0,1]$. What is the least degree $n$ for which the Lagrange error bound is less than $0.001$?

1. $5$
2. $7$
3. $6$
4. $4$

Question 82 · Multiple choice

The second-degree Maclaurin polynomial for $\ln(1+x)$ is used to approximate $\ln(1.2)$. Using $\left|f'''(x)\right|\le2$ for $0\le x\le0.2$, what is the Lagrange error bound?

1. $\frac{1}{750}$
2. $\frac{1}{375}$
3. $\frac{2}{375}$
4. $\frac{1}{125}$

Question 83 · Multiple choice

For a function $f$, $\left|f'''(x)\right|\le4$ for $0\le x\le0.1$. The second-degree Maclaurin polynomial gives $P_2(0.1)=1.105$. What is the Lagrange error bound for $|f(0.1)-1.105|$?

1. $\frac{1}{150}$
2. $\frac{1}{3000}$
3. $\frac{1}{600}$
4. $\frac{1}{1500}$

Question 84 · Multiple choice

The fourth-degree Maclaurin polynomial for $\cos x$ approximates $\cos1$. Using $\left|f^{(5)}(x)\right|\le1$, what is the Lagrange error bound?

1. $\frac{1}{24}$
2. $\frac{1}{720}$
3. $\frac{1}{120}$
4. $\frac{1}{5}$

Question 85 · Multiple choice

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n}$?

1. $-1\le x<1$
2. $-1<x\le1$
3. $-1<x<1$
4. $-1\le x\le1$

Question 86 · Multiple choice

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(x-2)^n}{3^n}$?

1. $-1\le x\le5$
2. $-1\le x<5$
3. $1<x<3$
4. $-1<x<5$

Question 87 · Multiple choice

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{n\,x^n}{4^n}$?

1. $-4\le x<4$
2. $-4<x<4$
3. $-4\le x\le4$
4. $-\frac14<x<\frac14$

Question 88 · Multiple choice

What is the radius of convergence of $\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}$?

1. $1$
2. $0$
3. $e$
4. $\infty$

Question 89 · Multiple choice

For which values of $x$ does $\displaystyle\sum_{n=0}^{\infty}n!\,x^n$ converge?

1. $-1<x<1$
2. $x=0$ only
3. All real $x$
4. $-1\le x\le1$

Question 90 · Multiple choice

What is the interval of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(x+1)^n}{n^2}$?

1. $-2\le x\le0$
2. $-2<x<0$
3. $-2\le x<0$
4. $0\le x\le2$

Question 91 · Multiple choice

What is the radius of convergence of $\displaystyle\sum_{n=1}^{\infty}\frac{(2x)^n}{\sqrt n}$?

1. $2$
2. $1$
3. $\frac{1}{2}$
4. $\frac{1}{4}$

Question 92 · Multiple choice

Which of the following gives the first four nonzero terms of the Maclaurin series for $e^{-x^3}$?

1. $1-x^3+\frac{x^6}{2}-\frac{x^9}{3}$
2. $1-x^3+\frac{x^6}{2}-\frac{x^9}{6}$
3. $1-x^3+x^6-x^9$
4. $1+x^3+\frac{x^6}{2}+\frac{x^9}{6}$

Question 93 · Multiple choice

What is the coefficient of $x^5$ in the Maclaurin series for $\sin x$?

1. $- \frac{1}{120}$
2. $\frac{1}{24}$
3. $\frac{1}{5}$
4. $\frac{1}{120}$

Question 94 · Multiple choice

Which gives the first four terms of the Taylor series for $\dfrac1x$ about $x=1$?

1. $1+(x-1)+(x-1)^2+(x-1)^3$
2. $1-(x-1)+\frac{(x-1)^2}{2}-\frac{(x-1)^3}{6}$
3. $1-(x-1)+(x-1)^2-(x-1)^3$
4. $1-x+x^2-x^3$

Question 95 · Multiple choice

A function has Maclaurin series beginning $1+3x+\dfrac92x^2+\dfrac92x^3+\cdots$. Which function could it be?

1. $e^{3x}$
2. $\frac{1}{1-3x}$
3. $(1+x)^3$
4. $3e^{x}-2$

Question 96 · Multiple choice

A function $f$ has $f^{(n)}(0)=\dfrac{n+1}{2^n}$ for all $n\ge0$. What is the coefficient of $x^3$ in its Maclaurin series?

1. $\frac{1}{12}$
2. $\frac{1}{2}$
3. $\frac{1}{6}$
4. $\frac{4}{3}$

Question 97 · Multiple choice

In the Taylor series for $e^x$ about $x=2$, what is the term containing $(x-2)^3$?

1. $\frac{1}{6}(x-2)^3$
2. $e^2(x-2)^3$
3. $\frac{e^2}{6}(x-2)^3$
4. $\frac{e^3}{6}(x-2)^3$

Question 98 · Multiple choice

What is the coefficient of $(x-1)^4$ in the Taylor series for $\ln x$ about $x=1$?

1. $\frac{1}{4}$
2. $- \frac{1}{4}$
3. $- \frac{1}{24}$
4. $-6$

Question 99 · Multiple choice

Which gives the first four nonzero terms of the Maclaurin series for $\dfrac{1}{1+4x^2}$?

1. $1+4x^2+16x^4+64x^6$
2. $1-4x+16x^2-64x^3$
3. $1-2x^2+4x^4-8x^6$
4. $1-4x^2+16x^4-64x^6$

Question 100 · Multiple choice

Integrating the series for $\dfrac{1}{1+x^2}$ term by term gives the Maclaurin series for $\arctan x$. What is the coefficient of $x^7$?

1. $\frac{1}{7}$
2. $- \frac{1}{6}$
3. $- \frac{1}{7}$
4. $- \frac{1}{5040}$

Question 101 · Multiple choice

Use the Maclaurin series for $\sin x$ to find $\displaystyle\lim_{x\to0}\frac{\sin x-x}{x^3}$.

1. $- \frac{1}{6}$
2. $\frac{1}{6}$
3. $0$
4. $-1$

Question 102 · Multiple choice

Differentiating $\dfrac{1}{1-x}=\displaystyle\sum_{n=0}^{\infty}x^n$ gives $\dfrac{1}{(1-x)^2}=\displaystyle\sum_{n=1}^{\infty}n\,x^{n-1}$ for $|x|<1$. What is $\displaystyle\sum_{n=1}^{\infty}\frac{n}{2^{n-1}}$?

1. $2$
2. $8$
3. The series diverges.
4. $4$

Question 103 · Multiple choice

What is the value of $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\left(\frac{\pi}{6}\right)^{2n+1}$?

1. $\frac{\sqrt{3}}{2}$
2. $\frac{1}{2}$
3. $\frac{\pi}{6}$
4. $0$

Question 104 · Multiple choice

What is the coefficient of $x^4$ in the Maclaurin series for $xe^{x}$?

1. $\frac{1}{24}$
2. $\frac{1}{4}$
3. $4$
4. $\frac{1}{6}$

Question 105 · Multiple choice

Using the first three nonzero terms of the Maclaurin series for $e^{-x^2}$, integrated term by term, what is the approximation of $\displaystyle\int_0^1e^{-x^2}\,dx$?

1. $\frac{2}{3}$
2. $\frac{23}{30}$
3. $\frac{5}{6}$
4. $\frac{7}{10}$

Question 106 · Free response (9 points)

Let $f(x)=e^{-2x}$.

1. (a) Write the third-degree Taylor polynomial $P_3(x)$ for $f$ about $x=0$. [2 points]
2. (b) Use $P_3$ to approximate $f(0.1)$. [1 point]
3. (c) Use the Lagrange error bound to show that $\left|f(0.1)-P_3(0.1)\right|<0.0001$. [2 points]
4. (d) Write the general term of the Maclaurin series for $f$, and give its interval of convergence. [2 points]
5. (e) Use the series to find $\displaystyle\lim_{x\to0}\frac{f(x)-1+2x}{x^2}$. [2 points]

Question 107 · Free response (9 points)

Let $g(x)=\displaystyle\sum_{n=1}^{\infty}\dfrac{(x-3)^n}{n\cdot2^n}$ for all $x$ where the series converges.

1. (a) Find the interval of convergence of the series. Justify your answer, including at the endpoints. [3 points]
2. (b) Write the first three nonzero terms of the series for $g'(x)$. [2 points]
3. (c) The series for $g'(x)$ is geometric. Find a closed-form expression for $g'(x)$ for $1<x<5$. [2 points]
4. (d) Does the series for $g(1)$ converge absolutely or conditionally? Explain. [2 points]

Question 108 · Free response (9 points)

This question is about several different series. Show the work that leads to each answer.

1. (a) Find the sum of $\displaystyle\sum_{n=1}^{\infty}\dfrac{6}{4^n}$. [1 point]
2. (b) Does $\displaystyle\sum_{n=1}^{\infty}\dfrac{3n^2}{n^2+4}$ converge or diverge? Give a reason. [1 point]
3. (c) Let $S_k=\displaystyle\sum_{n=1}^{k}\dfrac{1}{n^2+n}$. Write $S_k$ in terms of $k$, and find $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^2+n}$. [1 point]
4. (d) Use the ratio test to determine whether $\displaystyle\sum_{n=1}^{\infty}\dfrac{n^3}{3^n}$ converges. [2 points]
5. (e) Use the integral test to show that $\displaystyle\sum_{n=2}^{\infty}\dfrac{1}{n(\ln n)^2}$ converges. [2 points]
6. (f) Use the limit comparison test with $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{n^{3/2}}$ to determine whether $\displaystyle\sum_{n=1}^{\infty}\dfrac{\sqrt n}{n^2+1}$ converges. [1 point]
7. (g) The partial sum $S_5$ of $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n^3}$ is used to approximate the sum. Give a bound for the error. [1 point]