Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Sent to:
Search glass icon
  • Login
  • Textbooks
  • Ask our Educators
  • Study Tools
    Study Groups Bootcamps Quizzes AI Tutor iOS Student App Android Student App StudyParty
  • For Educators
    Become an educator Educator app for iPad Our educators
  • For Schools

  • Home
  • Textbooks
  • Calculus: Early Transcendentals
  • Infinite Sequences and Series

Calculus: Early Transcendentals

James Stewart

Chapter 11

Infinite Sequences and Series - all with Video Answers

Educators

JH

Section 2

Series

03:13

Problem 1

(a) What is the difference between a sequence and a series?
(b) What is a convergent series? What is a divergent series?

JH
J Hardin
Numerade Educator
02:17

Problem 2

Explain what it means to say that $ \sum_{n = 1}^{\infty} a_n = 5. $

JH
J Hardin
Numerade Educator
02:21

Problem 3

Calculate the sum of the series $ \sum_{n = 1}^{\infty} a_n $ whose partial sums are given.
$ s_n = 2 - 3(0.8)^n $

JH
J Hardin
Numerade Educator
02:13

Problem 4

Calculate the sum of the series $ \sum_{n = 1}^{\infty} a_n $ whose partial sums are given.
$ s_n = \frac {n^2 - 1}{4n^2 + 1} $

JH
J Hardin
Numerade Educator
09:23

Problem 5

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4 + n^2} $

JH
J Hardin
Numerade Educator
04:48

Problem 6

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt [3]{n}} $

JH
J Hardin
Numerade Educator
05:18

Problem 7

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \sin n $

JH
J Hardin
Numerade Educator
03:54

Problem 8

Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
$ \displaystyle \sum_{n = 1}^{\infty} \frac {(-1)^{n - 1}}{n!} $

JH
J Hardin
Numerade Educator
04:20

Problem 9

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {12}{(-5)^n} $

JH
J Hardin
Numerade Educator
03:08

Problem 10

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \cos n $

JH
J Hardin
Numerade Educator
04:29

Problem 11

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{\sqrt {n^2 + 4}} $

JH
J Hardin
Numerade Educator
05:44

Problem 12

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {7^{n + 1}}{10^n} $

JH
J Hardin
Numerade Educator
07:14

Problem 13

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {2}{n^2 - n} $

JH
J Hardin
Numerade Educator
07:39

Problem 14

Find at least 10 partial sums of the series. Graph both the sequence of terms and the sequence of partial sums on the same screen. Does it appear that the series is convergent or divergent? If it is convergent, find the sum. If it is divergent, explain why.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \sin \frac {1}{n} - \sin \frac {1}{n + 1} \right) $

JH
J Hardin
Numerade Educator
03:16

Problem 15

Let $ a_n = \frac {2n}{3n + 1} $
(a) Determine whether $ \left\{ a_n \right\} $ is convergent.
(b) Determine whether $ \sum_{n = 1}^{\infty} a_n $ is convergent.

JH
J Hardin
Numerade Educator
03:32

Problem 16

(a) Explain the difference between
$ \displaystyle \sum_{i = 1}^{n} a_i $ and $ \displaystyle \sum_{j = 1}^{n} a_j $
(b) Explain the difference between
$ \displaystyle \sum_{i = 1}^{n} a_i $ and $ \displaystyle \sum_{i = 1}^{n} a_j $

JH
J Hardin
Numerade Educator
02:28

Problem 17

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ 3 - 4 + \frac {16}{3} - {64}{9} + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
03:18

Problem 18

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ 4 + 3 + \frac {9}{4} + \frac {27}{16} + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
03:04

Problem 19

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ 10 - 2 + 0.4 - 0.08 + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
02:56

Problem 20

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ 2 + 0.5 + 0.125 + 0.03125 + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
02:32

Problem 21

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} 12(0.73)^{n-1} $

JH
J Hardin
Numerade Educator
02:04

Problem 22

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {5}{\pi^n} $

JH
J Hardin
Numerade Educator
03:45

Problem 23

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {(-3)^{n -1}}{4^n} $

JH
J Hardin
Numerade Educator
02:11

Problem 24

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 0}^{\infty} \frac {3^{n + 1}}{(-2)^n} $

JH
J Hardin
Numerade Educator
02:52

Problem 25

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^{2n}}{6^{n - 1}} $

JH
J Hardin
Numerade Educator
02:13

Problem 26

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {6 \cdot 2^{2n - 1}}{3^n} $

JH
J Hardin
Numerade Educator
01:15

Problem 27

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \frac {1}{3} + \frac {1}{6} + \frac {1}{9} + \frac {1}{12} + \frac {1}{15} + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
04:25

Problem 28

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \frac {1}{3} + \frac {2}{9} + \frac {1}{27} + \frac {2}{81} + \frac {1}{243} + \frac {2}{729} + \cdot \cdot \cdot $

JH
J Hardin
Numerade Educator
02:19

Problem 29

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2 + n}{1 - 2n} $

JH
J Hardin
Numerade Educator
02:23

Problem 30

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{k = 1}^{\infty} \frac {k^2}{k^2 - 2k + 5} $

JH
J Hardin
Numerade Educator
01:58

Problem 31

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} 3^{n + 1} 4^{-n} $

JH
J Hardin
Numerade Educator
03:33

Problem 32

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$$ \displaystyle \sum_{n = 1}^{\infty} [(-0.2)^n + (0.06)^{n - 1}] $$

JH
J Hardin
Numerade Educator
02:17

Problem 33

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{4 + e^{-n}} $

JH
J Hardin
Numerade Educator
02:26

Problem 34

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2^n + 4^n}{e^n} $

JH
J Hardin
Numerade Educator
03:24

Problem 35

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{k = 1}^{\infty} (\sin 100)^k $

JH
J Hardin
Numerade Educator
02:31

Problem 36

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{1 + (\frac {2}{3})^n} $

JH
J Hardin
Numerade Educator
02:48

Problem 37

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \ln \left( \frac {n^2 + 1}{2n^2 + 1} \right) $

JH
J Hardin
Numerade Educator
02:46

Problem 38

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$$ \displaystyle \sum_{k = 0}^{\infty} (\sqrt 2)^{-k} $$

JH
J Hardin
Numerade Educator
03:30

Problem 39

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \arctan n $

JH
J Hardin
Numerade Educator
02:35

Problem 40

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{5^n} + \frac {2}{n} \right) $

JH
J Hardin
Numerade Educator
06:21

Problem 41

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {1}{e^n} + \frac {1}{n(n + 1)} \right) $

JH
J Hardin
Numerade Educator
03:17

Problem 42

Determine whether the series is convergent or divergent. If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {e^n}{n^2} $

JH
J Hardin
Numerade Educator
06:27

Problem 43

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {2}{n^2 - 1} $

JH
J Hardin
Numerade Educator
04:12

Problem 44

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \ln \frac {n}{n + 1} $

JH
J Hardin
Numerade Educator
05:42

Problem 45

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {3}{n(n + 3)} $

JH
J Hardin
Numerade Educator
05:15

Problem 46

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 4}^{\infty} \left( \frac {1}{\sqrt n} - \frac {1}{\sqrt {n + 1}} \right) $

JH
J Hardin
Numerade Educator
04:28

Problem 47

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 1}^{\infty} \left( e^{1/n} - e^{1/(n + 1)} \right) $

JH
J Hardin
Numerade Educator
05:50

Problem 48

Determine whether the series is convergent or divergent by expressing $ s_n $ as a telescoping sum (as in Examples 8). If it is convergent, find its sum.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^3 - n} $

JH
J Hardin
Numerade Educator
05:55

Problem 49

Let $ x = 0.99999 . . . . $
(a) Do you think that $ x < 1 $ or $ x = 1? $
(b) Sum a geometric series to find the value of $ x. $
(c) How many decimal representations does the number 1 have?
(d) Which numbers have more than one decimal representation?

JH
J Hardin
Numerade Educator
02:58

Problem 50

A sequence of terms is defined by
$ a_1 = 1 $ and $ a_n = (5 - n) a_{n-1} $
Calculate $ \sum_{n = 1}^{\infty} a_n $.

JH
J Hardin
Numerade Educator
02:31

Problem 51

Express the number as a ratio of intergers.
$ 0. \overline 8 = 0.8888 . . . $

JH
J Hardin
Numerade Educator
04:16

Problem 52

Express the number as a ratio of integers.
$0 . \overline{46}=0.46464646 \ldots$

Mary Wakumoto
Mary Wakumoto
Numerade Educator
04:25

Problem 53

Express the number as a ratio of intergers.
$ 2. \overline {516} = 2.516516516 . . . $

JH
J Hardin
Numerade Educator
04:46

Problem 54

Express the number as a ratio of intergers.
$ 10.1 \overline {35} = 10.135353535 . . . $

JH
J Hardin
Numerade Educator
06:07

Problem 55

Express the number as a ratio of intergers.
$ 1.234 \overline {567} $

JH
J Hardin
Numerade Educator
07:01

Problem 56

Express the number as a ratio of intergers.
$ 5. \overline {71358} $

Chris Trentman
Chris Trentman
Numerade Educator
03:22

Problem 57

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 1}^{\infty} (-5)^n x^n $

JH
J Hardin
Numerade Educator
03:34

Problem 58

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 1}^{\infty} (x + 2)^n $

Chris Trentman
Chris Trentman
Numerade Educator
03:29

Problem 59

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} \frac {(x - 2)^n}{3^n} $

JH
J Hardin
Numerade Educator
03:54

Problem 60

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} (-4)^n (x - 5)^n $

JH
J Hardin
Numerade Educator
03:17

Problem 61

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} \frac {2^n}{x^n} $

JH
J Hardin
Numerade Educator
03:49

Problem 62

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} \frac {\sin^n x}{3^n} $

JH
J Hardin
Numerade Educator
03:23

Problem 63

Find the values of $ x $ for which the series converges. Find the sum of the series for those values of $ x. $
$ \displaystyle \sum_{n = 0}^{\infty} e ^{nx} $

JH
J Hardin
Numerade Educator
05:54

Problem 64

We have seen that the harmonic series is a divergent series whose terms approach 0. Show that
$ \displaystyle \sum_{n = 1}^{\infty} \ln \left( 1 + \frac {1}{n} \right) $
is another series with this property.

JH
J Hardin
Numerade Educator
03:53

Problem 65

Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using CAS to sum the series directly.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {3n^2 + 3n + 1}{(n^2 + n)^3} $

JH
J Hardin
Numerade Educator
07:50

Problem 66

Use the partial fraction command on your CAS to find a convenient expression for the partial sum, and then use this expression to find the sum of the series. Check your answer by using CAS to sum the series directly.
$ \displaystyle \sum_{n = 3}^{\infty} \frac {1}{n^5 - 5n^3 + 4n} $

JH
J Hardin
Numerade Educator
05:01

Problem 67

If the $ n $th partial sum of a series $ \sum_{n = 1}^{\infty} a_n $ is
$ s_n = \frac {n - 1}{n + 1} $
find $ a_n $ and $ \sum_{n = 1}^{\infty} a_n. $

JH
J Hardin
Numerade Educator
05:43

Problem 68

If the $ n $th partial sum of a series $ \sum_{n = 1}^{\infty} a_n $ is $ s_n = 3 - n2^{-n}, $ find $ a_n $ and $ \sum_{n = 1}^{\infty} a_n. $

JH
J Hardin
Numerade Educator
06:23

Problem 69

A doctor prescribes a 100-mg antibiotic tablet to be taken every eight hours. Just before each tablet is taken, $ 20% $ of the drug remains in the body.
(a) How much of the drug is in the body just after the second tablet is taken? After the third tablet?
(b) If $ Q_n $ is the quantity of the antibiotic in the body just after the $ n $th tablet is taken, find an equation that expresses $ Q_{n + 1} $ in terms of $ Q_n. $
(c) What quantity of the antibiotic remains in the body in the long run?

JH
J Hardin
Numerade Educator
09:09

Problem 70

A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by $ 90\% $ and the new dose in increase the concentration by 1.5 mg/L.
(a) What is the concentration after three doses?
(b) If $ C_n $ is the concentration after the $ n $th dose, find a formula for $ C_n $ as a function of $ n. $
(c) What is the limiting value of the concentration?

JH
J Hardin
Numerade Educator
05:21

Problem 71

A patient takes 150 mg of a drug at the same time every day. Just before each tablet is taken, $ 5% $ of the drug remains in the body.
(a) What quantity of the drug is in the body is in the body after the third tablet? After the $ n $th tablet?
(b) What quantity of the drug remains in the body in the long run?

JH
J Hardin
Numerade Educator
06:08

Problem 72

After injection of a does $ D $ of insulin, the concentration of insulin in patient's system decays exponentially and so it can be written as $ De^{-at}, $ where $ t $ represents time in hours and $ a $ is a positive constant.
(a) If a dose $ D $ is injected every $ T $ hours, write an expression for the sum of the residual concentrations just before the $ (n + 1) $st injection.
(b) Determine the limiting pre-injection concentration.
(c) If the concentration of insulin must always remain at or above a critical value $ C, $ determine a minimal dosage $ D $ in terms of $ C, a, $ and $ T. $

JH
J Hardin
Numerade Educator
04:38

Problem 73

When money is spent on goods and services, those who receive the money also spend some of it. The people receiving some of the twice-spent money will spend some of that, and so on. Economists call this chain reaction the multiplier effect. In a hypothetical isolated community, the local government begins the process by spending $ D $ dollars. Suppose that each recipient of spent money spends $ 100c% $ and saves $ 100s% $ of the money that he or she receives. The values $ c $ and $ s $ are called the marginal propensity to consume and the marginal propensity to save and, of course, $ c + s = 1. $
(a) Let $ S_n $ be the total spending that has been generated after $ n $ transactions. Find an equation for $ S_n. $
(b) Show that $ \lim_{n \to \infty} S_n = kD, $ where $ k = 1/s. $ The number $ k $ is called the multiplier. What is the multiplier if the marginal propensity to consume is $ 80%? $
Note: The federal government uses this principle to justify lending a large percentage of the money that they receive in deposits.

JH
J Hardin
Numerade Educator
12:38

Problem 74

A certain ball has the property that each time it falls from a height $ h $ onto a hard, level surface, it rebounds to a height $ rh, $ where $ 0 < r < 1. $ Suppose that the ball is dropped from an initial height of $ H $ meters.
(a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels.
(b) Calculate the total time that the ball travels. (Use the fact that the ball falls $ \frac {1}{2} gt^2 $ meters in $ t $ seconds.)
(c) Suppose that each time the ball strikes the surface with velocity $ v $ it rebounds with velocity $ -kv, $ where $ 0 < k < 1. $ How long will it take for the ball to come to rest?

JH
J Hardin
Numerade Educator
05:29

Problem 75

Find the value of $ c $ if
$ \displaystyle \sum_{n = 2}^{\infty} (1 + c)^{-n} = 2 $

JH
J Hardin
Numerade Educator
03:21

Problem 76

Find the value of c such that
$ \displaystyle \sum_{n = 0}^{\infty} e^{nc} = 10 $

JH
J Hardin
Numerade Educator
03:56

Problem 77

In Example 9 we showed that the harmonic series is divergent. Here we outline another method, making use of the fact that $ e^x > 1 + x $ for any $ x > 0. $ (See Exercise 4.3.84.)
If $ s_n $ is the $ n $th partial sum of the harmonic series, show that $ e^{x_n} > n + 1. $ Why does this imply that the harmonic series is divergent?

JH
J Hardin
Numerade Educator
07:11

Problem 78

Graph the curves $ y = x^n, 0 \le x \le 1, $ for $ n = 0, 1, 2, 3, 4, . . . $ on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example 8, that
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n(n + 1)} = 1 $

JH
J Hardin
Numerade Educator
12:15

Problem 79

The figure shows two circles $ C $ and $ D $ of radius 1 that touch at $ P. $ The line $ T $ is a common tangent line; $ C_1 $ is the circle that touches $ C, D, $ and $ T; C_2 $ is the circle that touches $ C, D, $ and $ C_1; C_3 $ is the circle that touches $ C, D, $ and $ C_2. $ This procedure can be continued indefinitely and produces an infinite sequence of circles $ \left \{C_n \right \}. $ Find an expression for the diameter of $ C_n $ and thus provide another geometric demonstration of Example 8.

JH
J Hardin
Numerade Educator
06:18

Problem 80

A right triangle $ ABC $ is given with $ \angle A = \theta $ and $ \mid AC \mid = b. $ $ CD $ is drawn perpendicular to $ AB, DE $ is drawn perpendicular to $ BC, EF \bot AB, $ and this process is continued indefinitely, as shown in the figure. Find the total length of all the perpendiculars
$ \mid CD \mid + \mid DE \mid + \mid EF \mid + \mid FG \mid + \cdot \cdot \cdot $
in terms of $ b $ and $ \theta. $

JH
J Hardin
Numerade Educator
04:19

Problem 81

What is wrong with the following calculation?
$ 0 = 0 + 0 + 0 + \cdot \cdot \cdot $
$ = (1 - 1) + (1 - 1) + (1 - 1) + \cdot \cdot \cdot $
$ = 1 - 1 + 1 - 1 + 1 - 1 + \cdot \cdot \cdot $
$ = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \cdot \cdot \cdot $
$ = 1 + 0 + 0 + 0 + \cdot \cdot \cdot = 1 $
(Guido Ubaldus thought that this proved the existence of God because "something has been created out of nothing.")

JH
J Hardin
Numerade Educator
03:59

Problem 82

Suppose that $ \sum_{n = 1}^{\infty} a_n \left( a_n \not= 0 \right) $ is known to be a convergent series. Prove that $ \sum_{n = 1}^{\infty} 1/a_n $ is a divergent series.

JH
J Hardin
Numerade Educator
03:13

Problem 83

Prove part (i) of Theorem 8.

JH
J Hardin
Numerade Educator
03:33

Problem 84

If $ \sum a_n $ is divergent and $ c \not= 0, $ show that $ \sum ca_n $ is divergent.

JH
J Hardin
Numerade Educator
02:30

Problem 85

If $ \sum a_n $ is convergent and $ \sum b_n $ is divergent, show that the series $ \sum \left( a_n + b_n \right) $ is divergent. [Hint: Argue by contradiction.]

JH
J Hardin
Numerade Educator
02:04

Problem 86

If $ \sum a_n $ and $ \sum b_n $ are both divergent, is $ \sum \left(a_n + b_n \right) $ necessarily divergent?

JH
J Hardin
Numerade Educator
03:47

Problem 87

Suppose that a series $ \sum a_n $ has positive terms and its partial sums $ s_n $ satisfy the inequality $ s_n \le 1000 $ for all $ n. $ Explain why $ \sum a_n $ must be convergent.

JH
J Hardin
Numerade Educator
12:37

Problem 88

The Fibonacci sequence was defined in Section 11.1 by the equations
$ f_1 = 1, f_2 = 1, f_n = f_{n -1} + f_{n - 2} n \ge 3 $
Show that each of the following statements is true.
(a) $ \frac {1}{f_{n - 1} f_{n + 1}} = \frac {1}{f_{n - 1} f_n} - \frac {1}{f_n f_{n + 1}} $
(b) $ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{f_{n - 1} f_{n + 1}} = 1 $
(c) $ \displaystyle \sum_{n = 2}^{\infty} \frac {f_n}{f_{n -1} f_{n + 1}} = 2 $

JH
J Hardin
Numerade Educator
08:30

Problem 89

The Cantor set, named after the German mathematician George Cantor (1845 - 1918), is constructed as follows. We start with the closed interval [0, 1] and remove the open interval $ \left( \frac {1}{3}, \frac {2}{3} \right). $ That leaves the two intervals $ \left[ 0, \frac {1}{3} \right] $ and $ \left[ \frac {2}{3}, 1 \right] $ and we remove the open middle third of each. Four intervals remain and given we remove the open middle third of each of them. We continue this procedure indefinitely, at each step removing the open middle third of every interval that remains from the preceding step. The Cantor set consists of the numbers that remain in [0, 1] after all those intervals have been removed.
(a) Show that the total length of all the intervals that are removed is 1. Despite that, the Cantor set contains infinitely many numbers. Give examples of some numbers in the Cantor set.
(b) The Sierpinski carpet is a two-dimensional counterpart of the Cantor set. It is constructed by removing the center one-ninth of a square of side 1, then removing the centers of the eight smaller remaining squares, and so on. (The figure shows the first three steps of the construction.) Show that the sum of the areas of the removed squares is 1. This implies that the Sierpinski carpet has area 0.

JH
J Hardin
Numerade Educator
07:11

Problem 90

(a) A sequence $ \left\{ a_n \right\} $ is defined recursively by the equation $ a_n = \frac {1}{2} \left(a_{n - 1} + a_{n - 2} \right) $ for $ n \ge 3, $ where $ a_1 $ and $ a_2 $ can be any real numbers. Experiment with various values of $ a_1 $ and $ a_2 $ and use your calculator to guess the limit of the sequence.
(b) Find $ \lim_{n \to \infty} a_n $ in terms of $ a_1 $ and $ a_2 $ by expressing $ a_{n + 1} - a_n $ in terms of $ a_2 - a_1 $ and summing a series.

JH
J Hardin
Numerade Educator
08:51

Problem 91

Consider the series $ \sum_{n = 1}^{\infty} n/(n + 1)!. $
(a) Find the partial sums $ s_1, s_2, s_3, $ and $ s_4. $ Do you recognize the denominators? Use the pattern to guess a formula for $ s_n. $
(b) Use mathematical induction to prove your guess.
(c) Show that the given infinite series is convergent, and find its sum.

JH
J Hardin
Numerade Educator
07:16

Problem 92

In the figure at the right there are infinitely many circles approaching the vertices of an equilateral triangle, each circle touching other circles and sides of the triangle. If the triangle has sides of length 1, find the total area occupied by the circles.

JH
J Hardin
Numerade Educator

Get 24/7 study help with our app

 

Available on iOS and Android

About
  • Our Story
  • Careers
  • Our Educators
  • Numerade Blog
Browse
  • Bootcamps
  • Books
  • Notes & Exams NEW
  • Topics
  • Test Prep
  • Ask Directory
  • Online Tutors
  • Tutors Near Me
Support
  • Help
  • Privacy Policy
  • Terms of Service
Get started