Calculus Single Variable

Brian E. Blank, Steven G. Krantz

Chapter 8

Infinite Series - all with Video Answers

Educators

Section 1

Series

Problem 1

In each of Exercises $1-20,$ evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{2 n^{2}-1+n^{3}}{5 n^{3}+n+2}
$$

Lucas Finney

Lucas Finney

Numerade Educator

Problem 2

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{3 n^{2}+n+4}{2 n^{3}+1}
$$

Lucas Finney

Numerade Educator

Problem 3

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{2^{n+1}+5}{2^{n}+3}
$$

Lucas Finney

Numerade Educator

Problem 4

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{3^{2 n}+2}{9^{n+1}+1}
$$

Lucas Finney

Numerade Educator

Problem 5

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{3^{n}+2 \cdot 5^{n}}{2^{n}+3 \cdot 5^{n}}
$$

Lucas Finney

Numerade Educator

Problem 6

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{2^{n}+5^{n}}{7^{n}}
$$

Lucas Finney

Numerade Educator

Problem 7

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{6 n+\cos (n)}{3 n+2-\sin \left(n^{2}\right)}
$$

Lucas Finney

Numerade Educator

Problem 8

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{6 n+\cos (n)}{3 n+2-\sin \left(n^{2}\right)}
$$

Lucas Finney

Numerade Educator

Problem 9

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{n 2^{n}+3^{n}+1}{n 2^{n}+3^{n}+2}
$$

Lucas Finney

Numerade Educator

Problem 10

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{\ln (n)}{n}
$$

Lucas Finney

Numerade Educator

Problem 11

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\frac{\ln ^{2}(n)}{\sqrt{n}}
$$

Lucas Finney

Numerade Educator

Problem 12

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=n e^{-2 n}
$$

Lucas Finney

Numerade Educator

Problem 13

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=n^{2} / e^{n}
$$

Lucas Finney

Numerade Educator

Problem 14

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\sum_{k=1}^{n} \frac{1}{k(k+1)}
$$

Lucas Finney

Numerade Educator

Problem 15

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\sqrt{n^{2}+3 n}-n
$$

Lucas Finney

Numerade Educator

Problem 16

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\arcsin \left(\frac{n}{n+1}\right)
$$

Lucas Finney

Numerade Educator

Problem 17

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\arctan (n)
$$

Lucas Finney

Numerade Educator

Problem 18

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=n \sin (1 / n)
$$

Lucas Finney

Numerade Educator

Problem 19

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\left(\frac{n+1}{n}\right)^{n}
$$

Lucas Finney

Numerade Educator

Problem 20

Evaluate $\lim _{n \rightarrow \infty} a_{n}$ for the given sequence $\left\{a_{n}\right\}$.
$$
a_{n}=\left(\frac{n-1}{n}\right)^{n}
$$

Lucas Finney

Numerade Educator

Problem 21

In each of Exercises $21-28,$ a series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty} 1 / n
$$

Lucas Finney

Numerade Educator

Problem 22

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty} 2^{n} / n !
$$

Lucas Finney

Numerade Educator

Problem 23

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty}\left(2^{-n}+1\right)
$$

Lucas Finney

Numerade Educator

Problem 24

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)
$$

Lucas Finney

Numerade Educator

Problem 25

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty} 2^{n} / 3^{n}
$$

Lucas Finney

Numerade Educator

Problem 26

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty} 1 / n^{2}
$$

Lucas Finney

Numerade Educator

Problem 27

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty}(-1)^{n+1} / n^{2}
$$

Lucas Finney

Numerade Educator

Problem 28

A series $\sum_{n=1}^{\infty} a_{n}$ is given. Calculate the first five partial sums of the series. That is, calculate $S_{N}=\sum_{n=1}^{N} a_{n}$ for $N=1,2,3,4,5$.
$$
\sum_{n=1}^{\infty} n^{2} / 2^{n}
$$

Lucas Finney

Numerade Educator

Problem 29

In each of Exercises $29-44,$ find the sum of the given series.
$$
\sum_{n=1}^{\infty}(3 / 7)^{n}
$$

Lucas Finney

Numerade Educator

Problem 30

Find the sum of the given series.
$$
\sum_{n=1}^{\infty}(2 / 3)^{2 n}
$$

Lucas Finney

Numerade Educator

Problem 31

Find the sum of the given series.
$$
\sum_{n=0}^{\infty}(-2 / 3)^{n}
$$

Lucas Finney

Numerade Educator

Problem 32

Find the sum of the given series.
$$
\sum_{n=1}^{\infty} 8^{-n}
$$

Lucas Finney

Numerade Educator

Problem 33

Find the sum of the given series.
$$
\sum_{n=0}^{\infty} 9(0.1)^{n}
$$

Lucas Finney

Numerade Educator

Problem 34

Find the sum of the given series.
$$
\sum_{n=4}^{\infty}(0.2)^{n}
$$

Lucas Finney

Numerade Educator

Problem 35

Find the sum of the given series.
$$
\sum_{n=-3}^{\infty}(1 / 5)^{n}
$$

Lucas Finney

Numerade Educator

Problem 36

Find the sum of the given series.
$$
\sum_{n=1}^{\infty} 7^{(-n / 3)}
$$

Lucas Finney

Numerade Educator

Problem 37

Find the sum of the given series.
$$
\sum_{n=2}^{\infty} \frac{1}{2^{n / 2}}
$$

Lucas Finney

Numerade Educator

Problem 38

Find the sum of the given series.
$$
\sum_{n=-1}^{\infty}(2 / 3)^{2 n+1}
$$

Lucas Finney

Numerade Educator

Problem 39

Find the sum of the given series.
$$
\sum_{n=3}^{\infty}(0.1)^{n} /(0.2)^{n+2}
$$

Lucas Finney

Numerade Educator

Problem 40

Find the sum of the given series.
$$
\sum_{n=3}^{\infty}\left(2^{-n}+3^{-n}\right)
$$

Lucas Finney

Numerade Educator

Problem 41

Find the sum of the given series.
$$
\sum_{n=1}^{\infty}\left(2^{-n} \cdot 3^{-n}+7^{-n}\right)
$$

Lucas Finney

Numerade Educator

Problem 42

Find the sum of the given series.
$$
\sum_{n=1}^{\infty}\left(5^{-n} \cdot 3^{-n} \cdot 4^{n}\right)
$$

Lucas Finney

Numerade Educator

Problem 43

Find the sum of the given series.
$$
\sum_{n=1}^{\infty} \frac{2^{n+1}}{5^{n-1}}
$$

Lucas Finney

Numerade Educator

Problem 44

Find the sum of the given series.
$$
\sum_{n=1}^{\infty} \frac{3^{n-1}}{4^{n+1}}
$$

Lucas Finney

Numerade Educator

Problem 45

In Exercises $45-50,$ write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.8888888 \ldots
$$

Lucas Finney

Numerade Educator

Problem 46

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.131313131313 \ldots
$$

William Semus

Numerade Educator

Problem 47

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.017017017 \ldots
$$

Lucas Finney

Numerade Educator

Problem 48

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.983983983 \ldots
$$

Lucas Finney

Numerade Educator

Problem 49

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.1221212121 \ldots
$$

Nick Johnson

Numerade Educator

Problem 50

Write each of the given repeating decimals as a constant times a geometric series (the geometric series will contain powers of 0.1 ). Use the formula for the sum of a geometric series to express the repeating decimal as a rational number.
$$
0.31323232 \ldots
$$

Nick Johnson

Numerade Educator

Problem 51

In Each of Exercises $51-56,$ the partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=2-1 / N^{2}
$$

Lucas Finney

Numerade Educator

Problem 52

The partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=(N+1) /(N+4)
$$

Lucas Finney

Numerade Educator

Problem 53

The partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=(3 N+2) /(N+1)
$$

Lucas Finney

Numerade Educator

Problem 54

The partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=2 N^{2} /\left(N^{2}+2\right)
$$

Lucas Finney

Numerade Educator

Problem 55

The partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=(3 N+1) /(2 N+4)
$$

Lucas Finney

Numerade Educator

Problem 56

The partial sum $S_{N}=\sum_{n=1}^{N} a_{n}$ of an infinite series $\sum_{n=1}^{\infty} a_{n}$ is given. Determine the value of the infinite series.
$$
S_{N}=2-2^{N+1} / 3^{N+2}
$$

Lucas Finney

Numerade Educator

Problem 57

In each of Exercises $57-62$, prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty}(1.01)^{n}
$$

Nick Johnson

Numerade Educator

Problem 58

Prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty} \frac{n}{n+1}
$$

Nick Johnson

Numerade Educator

Problem 59

Prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty} \frac{n}{2 n+3}
$$

Nick Johnson

Numerade Educator

Problem 60

Prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty} \frac{n}{\sqrt{n^{2}+1}}
$$

Nick Johnson

Numerade Educator

Problem 61

Prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty} \frac{4^{n}}{4^{n}+2^{n}}
$$

Nick Johnson

Numerade Educator

Problem 62

Prove that the given series diverges by showing that the $N^{\text {th }}$ partial sum satisfies $S_{N} \geq k \cdot N$ for some positive constant $k$.
$$
\sum_{n=1}^{\infty} \frac{n+10}{10 n}
$$

Nick Johnson

Numerade Educator

Problem 63

In each of Exercises $63-68,$ calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)
$$

Nick Johnson

Numerade Educator

Problem 64

Calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}\left(\frac{n+1}{n+2}-\frac{n}{n+1}\right)
$$

Nick Johnson

Numerade Educator

Problem 65

Calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}\left(\frac{2 n-2}{n^{3}}-\frac{2 n}{(n+1)^{3}}\right)
$$

Nick Johnson

Numerade Educator

Problem 66

Calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}\left(\frac{1}{n^{2}}-\frac{1}{(n+1)^{2}}\right)
$$

Nick Johnson

Numerade Educator

Problem 67

Calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)
$$

Nick Johnson

Numerade Educator

Problem 68

Calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty}(\arctan (n+1)-\arctan (n))
$$

Nick Johnson

Numerade Educator

Problem 69

In each of Exercises $69-72,$ use partial fractions to calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty} \frac{1}{n(n+2)}
$$

Nick Johnson

Numerade Educator

Problem 70

Use partial fractions to calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty} \frac{1}{n(n+4)}
$$

Nick Johnson

Numerade Educator

Problem 71

Use partial fractions to calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty} \frac{1}{(2 n+1)(2 n+3)}
$$

Nick Johnson

Numerade Educator

Problem 72

Use partial fractions to calculate the $N^{\text {th }}$ partial sum $S_{N}$ of the given series in closed form. Sum the series by finding $\lim _{N \rightarrow \infty} S_{N}$.
$$
\sum_{n=1}^{\infty} \frac{2 n+1}{\left(n^{2}+n\right)^{2}}
$$

Nick Johnson

Numerade Educator

Problem 73

Calculate the $N^{\mathrm{th}}$ partial sum of $\sum_{n=1}^{\infty} \ln \left(\frac{n}{n+1}\right)$. Show that the series is divergent.

Nick Johnson

Numerade Educator

Problem 74

Show that $\sum_{n=2}^{N} \ln \left(1-\frac{1}{n^{2}}\right)=\ln ((N-1) !)+\ln ((N+1) !)-2 \ln (N !)-\ln (2)$
and use this formula to sum $\sum_{n=2}^{\infty} \ln \left(1-\frac{1}{n^{2}}\right)$.

Nick Johnson

Numerade Educator

Problem 75

Suppose that $r$ is a constant greater than $1 .$ Calculate the $N^{\text {th }}$ partial sum of $\sum_{n=1}^{\infty} \frac{r^{n}}{\left(r^{n}-1\right)\left(r^{n+1}-1\right)},$ and use the formula to evaluate the infinite series.

Nick Johnson

Numerade Educator

Problem 76

The Fibonacci sequence $\left\{f_{n}\right\}$ is recursively defined by $f_{0}=f_{1}=1$ and $f_{n}=f_{n-1}+f_{n-2}$ for $n \geq 2 .$ Show that
$$
\frac{1}{f_{n} f_{n+2}}=\frac{1}{f_{n} f_{n+1}}-\frac{1}{f_{n+1} f_{n+2}}
$$
and use this formula to sum $\sum_{n=0}^{\infty} 1 /\left(f_{n} f_{n+2}\right)$.

Nick Johnson

Numerade Educator

Problem 77

It is known that $\sum_{n=1}^{\infty} 1 / n^{2}=\pi^{2} / 6 .$ What are the values of the convergent series
$$
1 / 2^{2}+1 / 4^{2}+1 / 6^{2}+1 / 8^{2}+\cdots
$$
and
$$
1+1 / 3^{2}+1 / 5^{2}+1 / 7^{2}+\cdots ?
$$

Nick Johnson

Numerade Educator

Problem 78

It is known that $\sum_{n=1}^{\infty} \frac{n}{2^{n}}=2$ and $\sum_{n=1}^{\infty} \frac{1}{n 2^{n}}=\ln (2)$.
Supposing that $a, b,$ and $c$ are constants, evaluate $\sum_{n=1}^{\infty} \frac{a n^{2}+c n+b}{n 2^{n}}$

Nick Johnson

Numerade Educator

Problem 79

It is known that $\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{1}{6} \pi^{2}$ and $\sum_{n=1}^{\infty} \frac{1}{n^{4}}=\frac{1}{90} \pi^{4}$. Let
$f(x, y)=\sum_{n=1}^{\infty} \frac{x+y \cdot(\pi n)^{2}}{n^{4}} .$ Describe the solution set $\left\{(x, y) \in \mathbb{R}^{2}: f(x, y)=0\right\}$.

Nick Johnson

Numerade Educator

Problem 80

The infinite series $\sum_{n=1}^{\infty} n r^{n-1}$ arises in genetics and other fields. Notice that the partial $S_{N}=\sum_{n=1}^{N} n r^{n-1}$ is equal to $\frac{d}{d r} \sum_{n=1}^{N} r^{n} .$ Use this observation to derive a closed form expression for $S_{N}$. Show that $\sum_{n=1}^{\infty} n r^{n-1}$ converges for $|r|<1$, and determine its value.

Nick Johnson

Numerade Educator

Problem 81

Prove that $\sum_{n=1}^{\infty} \ln (1 / n)$ diverges.

Nick Johnson

Numerade Educator

Problem 82

Use the inequality $x / 2<\ln (1+x)$ for $x \in(0,1)$ to prove that $\sum_{n=1}^{\infty} \ln (1+1 / n)$ diverges.

Nick Johnson

Numerade Educator

Problem 83

A tournament ping-pong ball bounces to $2 / 3$ of its original height when it is dropped from a height of $25 \mathrm{~cm}$ or less onto a hard surface. How far will the ball travel if dropped from a height of $20 \mathrm{~cm}$ and allowed to bounce forever?

Nick Johnson

Numerade Educator

Problem 84

Suppose that every dollar that we spend gives rise (through wages, profits, etc.) to 90 cents for someone else to spend. That 90 cents will generate a further 81 cents for spending, and so on. How much spending will result from the purchase of a $\$ 16,000$ automobile, the car included? (This phenomenon is known as the multiplier effect.)

Nick Johnson

Numerade Educator

Problem 85

Suppose that the national savings rate is $5 \% .$ Of every dollar earned, $5 \phi$ is saved and $95 \phi$ is spent. Of the $95 \phi$ that goes into the hands of others, $5 \%$ will be saved and $95 \%$ spent, and so on. How much spending will be generated by a 100 billion dollar $\left(\$ 10^{11}\right)$ tax cut?

Nick Johnson

Numerade Educator

Problem 86

A heart patient must take a $0.5 \mathrm{mg}$ daily dose of a medication. Each day, his body eliminates $90 \%$ of the medication present. The amount $S_{N}$ of the medicine that is present after $N$ days is a partial sum of which infinite series? Approximately what amount of the medicine is maintained in the patient's body after a long period of treatment?

Nick Johnson

Numerade Educator

Problem 87

A particular pollutant breaks down in water as follows: If the mass of the pollutant at the beginning of the time interval $[t, t+T]$ is $M,$ then the mass is reduced to $M e^{-k T}$ by the end of the interval. Suppose that at intervals $0, T$, $2 T, 3 T, \ldots$ a factory discharges an amount $M_{0}$ of the pollutant into a holding tank containing water. After a long period, the mixture from this tank is fed into a river. If the quantity of pollutant released into the river is required to be no greater than $Q,$ then, in terms of $k, T$ and $Q$, how large can $M_{0}$ be if the factory is compliant?

Nick Johnson

Numerade Educator

Problem 88

Sketch the graphs of $y=x^{n}, n=1,2,3, \ldots$ in one viewing window $[0,1] \times[0,1] .$ Find the area between two consecutive graphs $y=x^{n-1}$ and $y=x^{n}$. Use your calculation to calculate $\sum_{n=1}^{\infty} 1 /(n \cdot(n+1))$ by a geometric argument. If $r$ is a fixed number in $(0,1),$ then $\sum_{n=1}^{\infty} r^{n}$ converges.

Nick Johnson

Numerade Educator

Problem 89

If $r$ is a fixed number in $(0,1),$ then $\sum_{n=1}^{\infty} r^{n}$ converges. However, if $\left\{r_{n}\right\}$ is a sequence of numbers in $(0,1),$ then $\sum_{n=1}^{\infty} r_{n}^{n}$ may diverge. Prove the divergence of this series for $r_{n}=1-1 / n$.

Nick Johnson

Numerade Educator

Problem 90

Show that
$$
2 \cdot 3 \cdot 6 \cdot 8 \cdots(2 N)=N ! 2^{N}
$$

Nick Johnson

Numerade Educator

Problem 91

Show that
$$
1 \cdot 3 \cdot 5 \cdot 7 \cdots(2 N-3)(2 N-1)=\frac{(2 N) !}{N ! 2^{N}}
$$

Nick Johnson

Numerade Educator

Problem 92

Mr. Woodman has an unlimited supply of 1 inch long, $3 / 16$ inch thick dominoes. He stacks $N$ of them so that for each $2 \leq n \leq N,$ the $n^{\text {th }}$ domino, counting from the bottom of the stack, protrudes $1 /(2(N-n+1))$ inch over the end of the $(n-1)^{\text {st }}$ domino (see Figure 1 ). Show that the center of mass of Mr. Woodman's tower of dominoes lies over the bottom domino (and so the stack will not fall). Deduce that by using a sufficiently large number $N$, Mr. Woodman can make his stack span, from left to right, any given distance. About how many dominoes would it take to span the 10 foot length of his playroom? How high would the tower be?

James Chok

Numerade Educator

Problem 93

In each of Exercises $93-97$, a convergent series is given. Estimate the value $\ell$ of the series by calculating its partial sums $S_{N}$ for $N=1,2,3, \ldots$ Round your evaluations to four decimal places and stop when three consecutive rounded partial sums agree. (This procedure does not ensure that the last partial sum calculated agrees with $\ell$ to four decimal places. The error that results when a partial sum is used to approximate an infinite series is called a truncation error. Methods of estimating truncation errors will be discussed in later sections.)
$$
\sum_{n=1}^{\infty} \frac{e^{-n}}{n}
$$

Nick Johnson

Numerade Educator

Problem 94

A convergent series is given. Estimate the value $\ell$ of the series by calculating its partial sums $S_{N}$ for $N=1,2,3, \ldots$ Round your evaluations to four decimal places and stop when three consecutive rounded partial sums agree. (This procedure does not ensure that the last partial sum calculated agrees with $\ell$ to four decimal places. The error that results when a partial sum is used to approximate an infinite series is called a truncation error. Methods of estimating truncation errors will be discussed in later sections.)
$$
\sum_{n=1}^{\infty} \frac{e^{n}}{n !}
$$

Nick Johnson

Numerade Educator

Problem 95

A convergent series is given. Estimate the value $\ell$ of the series by calculating its partial sums $S_{N}$ for $N=1,2,3, \ldots$ Round your evaluations to four decimal places and stop when three consecutive rounded partial sums agree. (This procedure does not ensure that the last partial sum calculated agrees with $\ell$ to four decimal places. The error that results when a partial sum is used to approximate an infinite series is called a truncation error. Methods of estimating truncation errors will be discussed in later sections.)
$$
\sum_{n=1}^{\infty}\left(\frac{1.1}{n}\right)^{n}
$$

Nick Johnson

Numerade Educator

Problem 96

A convergent series is given. Estimate the value $\ell$ of the series by calculating its partial sums $S_{N}$ for $N=1,2,3, \ldots$ Round your evaluations to four decimal places and stop when three consecutive rounded partial sums agree. (This procedure does not ensure that the last partial sum calculated agrees with $\ell$ to four decimal places. The error that results when a partial sum is used to approximate an infinite series is called a truncation error. Methods of estimating truncation errors will be discussed in later sections.)
$$
\sum_{n=1}^{\infty} \frac{2^{n}+n^{2}}{10^{n}}
$$

Nick Johnson

Numerade Educator

Problem 97

A convergent series is given. Estimate the value $\ell$ of the series by calculating its partial sums $S_{N}$ for $N=1,2,3, \ldots$ Round your evaluations to four decimal places and stop when three consecutive rounded partial sums agree. (This procedure does not ensure that the last partial sum calculated agrees with $\ell$ to four decimal places. The error that results when a partial sum is used to approximate an infinite series is called a truncation error. Methods of estimating truncation errors will be discussed in later sections.)
$$
\sum_{n=1}^{\infty} \frac{2 n+1}{n^{5}+n+1}
$$

Nick Johnson

Numerade Educator

Problem 98

Let $a$ and $b$ be positive numbers, and set $r=1 /\left(1+a^{-b}\right)$. Observe that $0<r<1,$ and therefore $\sum_{n=1}^{\infty} r^{n}$ is convergent. Let $S$ denote the value of this infinite series.
a. Let $\tau(N)=\sum_{n=N+1}^{\infty} r^{n} .$ Show that
$$
\tau(N)=\frac{a^{b(N+1)}}{\left(a^{b+1)^{N}}\right.}
$$
b. For what constants $\alpha$ and $\beta$ is $\ln (\tau(N))=\alpha \ln (a)-\beta \ln$
$\left(a^{b}+1\right) ?$
c. Calculate $\ln (\tau(N))$ for $a=10, b=50,$ and $N=10^{6} .$ By about how much does the millionth partial sum $S_{10^{6}}$ of
$$
\sum_{n=1}^{\infty}\left(\frac{1}{1+10^{-50}}\right)^{n}
$$
differ from the full sum? (The difference is very large. The number $N$ of terms that must be summed for $S-S_{N}$ to be less than 0.1 is around $\left.1.17 \times 10^{52} .\right)$

Lucas Gagne

Numerade Educator