Calculus

Dale Varberg, Edwin Purcell deceased, Steve Rigdon

Chapter 9

Infinite Series - all with Video Answers

Educators

Section 1

Infinite Sequences

Problem 1

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{n}{3 n-1}$$

Wendi Zhao

Wendi Zhao

Numerade Educator

Problem 2

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{3 n+2}{n+1}$$

Wendi Zhao

Numerade Educator

Problem 3

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{4 n^{2}+2}{n^{2}+3 n-1}$$

Wendi Zhao

Numerade Educator

Problem 4

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{3 n^{2}+2}{2 n-1}$$

Wendi Zhao

Numerade Educator

Problem 5

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{n^{3}+3 n^{2}+3 n}{(n+1)^{3}}$$

Wendi Zhao

Numerade Educator

Problem 6

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{\sqrt{3 n^{2}+2}}{2 n+1}$$

Wendi Zhao

Numerade Educator

Problem 7

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=(-1)^{n} \frac{n}{n+2}$$

Wendi Zhao

Numerade Educator

Problem 8

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{n \cos (n \pi)}{2 n-1}$$

Wendi Zhao

Numerade Educator

Problem 9

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{\cos (n \pi)}{n}$$

Wendi Zhao

Numerade Educator

Problem 10

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=e^{-n} \sin n$$

Wendi Zhao

Numerade Educator

Problem 11

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{e^{2 n}}{n^{2}+3 n-1}$$

Wendi Zhao

Numerade Educator

Problem 12

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{e^{2 n}}{4^{n}}$$

Wendi Zhao

Numerade Educator

Problem 13

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{(-\pi)^{n}}{5^{n}}$$

Wendi Zhao

Numerade Educator

Problem 14

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\left(\frac{1}{4}\right)^{n}+3^{n / 2}$$

Wendi Zhao

Numerade Educator

Problem 15

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=2+(0.99)^{n}$$

Wendi Zhao

Numerade Educator

Problem 16

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{n^{100}}{e^{n}}$$

Wendi Zhao

Numerade Educator

Problem 17

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{\ln n}{\sqrt{n}}$$

Wendi Zhao

Numerade Educator

Problem 18

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\frac{\ln (1 / n)}{\sqrt{2 n}}$$

Wendi Zhao

Numerade Educator

Problem 19

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=\left(1+\frac{2}{n}\right)^{n / 2}$$

Wendi Zhao

Numerade Educator

Problem 20

An explicit formula for $a_{n}$ is given. Write the first five terms of $\left\{a_{n}\right\},$ determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$a_{n}=(2 n)^{1 / 2 n}$$

Wendi Zhao

Numerade Educator

Problem 21

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \dots$$

Wendi Zhao

Numerade Educator

Problem 22

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$\frac{1}{2^{2}}, \frac{2}{2^{3}}, \frac{3}{2^{4}}, \frac{4}{2^{5}}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 23

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$-1, \frac{2}{3},-\frac{3}{5}, \frac{4}{7},-\frac{5}{9}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 24

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$1, \frac{1}{1-\frac{1}{2}}, \frac{1}{1-\frac{2}{3}}, \frac{1}{1-\frac{3}{4}}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 25

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$1, \frac{2}{2^{2}-1^{2}}, \frac{3}{3^{2}-2^{2}}, \frac{4}{4^{2}-3^{2}}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 26

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$\frac{1}{2-\frac{1}{2}}, \frac{2}{3-\frac{1}{3}}, \frac{3}{4-\frac{1}{4}}, \frac{4}{5-\frac{1}{5}}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 27

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$\sin 1,2 \sin \frac{1}{2}, 3 \sin \frac{1}{3}, 4 \sin \frac{1}{4}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 28

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$-\frac{1}{3}, \frac{4}{9},-\frac{9}{27}, \frac{16}{81}, \dots$$

Wendi Zhao

Numerade Educator

Problem 29

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$2,1, \frac{2^{3}}{3^{2}}, \frac{2^{4}}{4^{2}}, \frac{2^{5}}{5^{2}}, \dots$$

Wendi Zhao

Numerade Educator

Problem 30

Find an explicit formula $a_{n}=$ for each sequence, determine whether the sequence converges or diverges, and, if it converges, find $\lim _{n \rightarrow \infty} a_{n}$
$$1-\frac{1}{2}, \frac{1}{2}-\frac{1}{3}, \frac{1}{3}-\frac{1}{4}, \frac{1}{4}-\frac{1}{5}, \ldots$$

Wendi Zhao

Numerade Educator

Problem 31

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{n}=\frac{4 n-3}{2^{n}}$$

Wendi Zhao

Numerade Educator

Problem 32

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{n}=\frac{n}{n+1}\left(2-\frac{1}{n^{2}}\right)$$

Wendi Zhao

Numerade Educator

Problem 33

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{n}=\left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right) \cdots\left(1-\frac{1}{n^{2}}\right), n \geq 2$$

Wendi Zhao

Numerade Educator

Problem 34

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{n}=1+\frac{1}{2 !}+\frac{1}{3 !}+\dots+\frac{1}{n !}$$

Wendi Zhao

Numerade Educator

Problem 35

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{1}=1, a_{n+1}=1+\frac{1}{2} a_{n}$$

Wendi Zhao

Numerade Educator

Problem 36

Write the first four terms of the sequence $\left\{a_{n}\right\}$ Then use Theorem D to show that the sequence converges.
$$a_{1}=2, a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{2}{a_{n}}\right)$$

Wendi Zhao

Numerade Educator

Problem 37

Assuming that $u_{1}=\sqrt{3}$ and $u_{n+1}=\sqrt{3+u_{n}}$ determine a convergent sequence, find $\lim _{n \rightarrow \infty} u_{n}$ to four decimal places.

Wendi Zhao

Numerade Educator

Problem 38

Show that $\left\{u_{n}\right\}$ of Problem 37 is bounded above and increasing. Conclude from Theorem D that $\left\{u_{n}\right\}$ converges.

Wendi Zhao

Numerade Educator

Problem 39

Find $\lim _{n \rightarrow \infty} u_{n}$ of Problem 37 algebraically. Hint: Let $u=\lim _{n \rightarrow \infty} u_{n} .$ Then, since $u_{n+1}=\sqrt{3+u_{n}}, u=\sqrt{3+u} .$ Now
square both sides and solve for $u$

Wendi Zhao

Numerade Educator

Problem 40

Use the technique of Problem 39 to find $\lim _{n \rightarrow \infty} a_{n}$ of Problem 36

Wendi Zhao

Numerade Educator

Problem 41

Assuming that $u_{1}=0$ and $u_{n+1}=1.1^{u_{e}}$ determine a convergent sequence, find $\lim _{n \rightarrow \infty} u_{n}$ to four decimal places.

Wendi Zhao

Numerade Educator

Problem 42

Show that $\left\{u_{n}\right\}$ of Problem 41 is increasing and bounded above by 2.

Wendi Zhao

Numerade Educator

Problem 43

Find $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \frac{k}{n}\right) \frac{1}{n}$$

Wendi Zhao

Numerade Educator

Problem 44

Show that $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left[\frac{1}{1+(k / n)^{2}}\right] \frac{1}{n}=\frac{\pi}{4}$$

Wendi Zhao

Numerade Educator

Problem 45

Using the definition of limit, prove that $\lim _{n \rightarrow \infty} n /(n+1)$ $=1 ;$ that is, for a given $\varepsilon>0,$ find $N$ such that $n \geq N \Rightarrow|n /(n+1)-1|<\varepsilon$

Wendi Zhao

Numerade Educator

Problem 46

$$\text { As in Problem } 45, \text { prove that } \lim _{n \rightarrow \infty} n /\left(n^{2}+1\right)=0$$

Wendi Zhao

Numerade Educator

Problem 47

Let $S=\left\{x: x \text { is rational and } x^{2}<2\right\}$. Convince yourself that $S$ does not have a least upper bound in the rational numbers, but does have such a bound in the real numbers. In other words, the sequence of rational numbers $1,1.4,1.41,1.414, \ldots,$ has no limit within the rational numbers.

Leon Druch

Numerade Educator

Problem 48

The completeness property of the real numbers says that for every set of real numbers that is bounded above, there exists a real number that is a least upper bound for the set. This property is usually taken as an axiom for the real numbers. Prove Theorem D using this property.

Faizanullah Kazmi

Faizanullah Kazmi

Numerade Educator

Problem 49

Prove that if $\lim _{n \rightarrow \infty} a_{n}=0$ and $\left\{b_{n}\right\}$ is bounded then $\lim _{n \rightarrow \infty} a_{n} b_{n}=0$

JH

J Hardin

Numerade Educator

Problem 50

Prove that if $\left\{a_{n}\right\}$ converges and $\left\{b_{n}\right\}$ diverges then $\left\{a_{n}+b_{n}\right\}$ diverges.

Wendi Zhao

Numerade Educator

Problem 51

If $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ both diverge, does it follow that $\left\{a_{n}+b_{n}\right\}$ diverges?

Wendi Zhao

Numerade Educator

Problem 52

A famous sequence $\left\{f_{n}\right\}$, called the Fibonacci Sequence after Leonardo Fibonacci, who introduced it around A.D. 1200, is defined by the recursion formula $$f_{1}=f_{2}=1, \quad f_{n+2}=f_{n+1}+f_{n}$$ (a) Find $f_{3}$ through $f_{10}$
(b) Let $\phi=\frac{1}{2}(1+\sqrt{5}) \approx 1.618034 .$ The Greeks called this number the golden ratio, claiming that a rectangle whose dimensions were in this ratio was "perfect." It can be shown that $$\begin{aligned}
f_{n} &=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right] \\
&=\frac{1}{\sqrt{5}}\left[\phi^{n}-(-1)^{n} \phi^{-n}\right]
\end{aligned}$$ Check that this gives the right result for $n=1$ and $n=2$ The general result can be proved by induction (it is a nice challenge). More in line with this section, use this explicit formula to prove that $\lim _{n \rightarrow \infty} f_{n+1} / f_{n}=\phi$
(c) Using the limit just proved, show that $\phi$ satisfies the equation $x^{2}-x-1=0 .$ Then, in another interesting twist, use the Quadratic Formula to show that the two roots of this equation are $\phi$ and $-1 / \phi,$ two numbers that occur in the explicit formula for $f_{n}$

Carson Merrill

Numerade Educator

Problem 53

Consider an equilateral triangle containing $1+2+$ $3+\cdots+n=n(n+1) / 2$ circles, each of diameter 1 and stacked as indicated in Figure 4 for the case $n=4$. Find $\lim _{n \rightarrow \infty} A_{n} / B_{n},$ where $A_{n}$ is the total area of the circles, and $B_{n}$ is the area of the triangle.

Mohamed Raafat Mohamed

Mohamed Raafat Mohamed

Numerade Educator

Problem 54

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}$$

Wendi Zhao

Numerade Educator

Problem 55

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{2 n}\right)^{n}$$

Wendi Zhao

Numerade Educator

Problem 56

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(1+\frac{1}{n^{2}}\right)^{n}$$

Wendi Zhao

Numerade Educator

Problem 57

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(\frac{n-1}{n+1}\right)^{n}$$

Wendi Zhao

Numerade Educator

Problem 58

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(\frac{2+n^{2}}{3+n^{2}}\right)^{n}$$

Wendi Zhao

Numerade Educator

Problem 59

Use the fact that $\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow 0^{+}} f(\frac{1}{x}$ to find the limits.
$$\lim _{n \rightarrow \infty}\left(\begin{array}{l} 2+n^{2} \\ 3+n^{2} \end{array}\right)^{n^{2}}$$

Wendi Zhao

Numerade Educator