Calculus: Early Transcendentals

James Stewart

Chapter 11

Infinite Sequences and Series - all with Video Answers

Educators

JH

EI

Section 3

The Integral Test and Estimates of Sums

Problem 1

1. Draw a picture to show that
$ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^{1.3}} < \int^{\infty}_1 \frac {1}{x^{1.3}} dx $
What can you conclude about the series?

Clayton Craig

Clayton Craig

Numerade Educator

Problem 2

Suppose $ f $ is a continuous positive decreasing function for $ x \ge 1 $ and $ a_n = f(n). $ By drawing a picture, rank the following three quantities in increasing order.

$ \int^6_1 f(x) dx \displaystyle \sum_{i = 1}^{5} a_i \displaystyle \sum_{i = 2}^6 a_i $

Clayton Craig

Numerade Educator

Problem 3

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} n^{-3} $

Clayton Craig

Numerade Educator

Problem 4

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} n^{-0.3} $

Clayton Craig

Numerade Educator

Problem 5

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {2}{5n - 1} $

Clayton Craig

Numerade Educator

Problem 6

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\left(3n - 1 \right)^34} $

Clayton Craig

Numerade Educator

Problem 7

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{n^2 + 1} $

Nick Johnson

Numerade Educator

Problem 8

Use the Integral Test to determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} n^2 e^{-n^3} $

Madi Sousa

Numerade Educator

Problem 9

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^{\sqrt 2}} $

Clayton Craig

Numerade Educator

Problem 10

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 3}^{\infty} n^{-0.9999} $

Madi Sousa

Numerade Educator

Problem 11

Determine whether the series is convergent or divergent.
$ 1 + \frac {1}{8} + \frac {1}{27} + \frac {1}{64} + \frac {1}{125} + \cdot \cdot \cdot $

Madi Sousa

Numerade Educator

Problem 12

Determine whether the series is convergent or divergent.
$ \frac {1}{5} + \frac {1}{7} + \frac {1}{9} + \frac {1}{11} + \frac {1}{13} + \cdot \cdot \cdot $

Chris Trentman

Numerade Educator

Problem 13

Determine whether the series is convergent or divergent.
$ \frac {1}{3} + \frac {1}{7} + \frac {1}{11} + \frac {1}{15} + \frac {1}{19} + \cdot \cdot \cdot $

Clayton Craig

Numerade Educator

Problem 14

Determine whether the series is convergent or divergent.
$ 1 + \frac {1}{2 \sqrt 2} + \frac {1}{ 3 \sqrt 3} + \frac {1}{4 \sqrt 4} + \frac {1}{5 \sqrt 5} + \cdot \cdot \cdot $

Clayton Craig

Numerade Educator

Problem 15

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt n + 4}{n^2} $

Clayton Craig

Numerade Educator

Problem 16

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\sqrt n}{1 + n^{3/2}} $

JH

J Hardin

Numerade Educator

Problem 17

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 4} $

Carson Merrill

Numerade Educator

Problem 18

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + 2n + 2} $

Carson Merrill

Numerade Educator

Problem 19

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n^3}{n^4 + 4} $

Carson Merrill

Numerade Educator

Problem 20

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 3}^{\infty} \frac {3n - 4}{n^2 - 2n} $

Clayton Craig

Numerade Educator

Problem 21

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n \ln n} $

Clayton Craig

Numerade Educator

Problem 22

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {\ln n}{n^2} $

Clayton Craig

Numerade Educator

Problem 23

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{k = 1}^{\infty} ke^{-k} $

Clayton Craig

Numerade Educator

Problem 24

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{k = 1}^{\infty} ke^{-k^2} $

Clayton Craig

Numerade Educator

Problem 25

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2 + n^3} $

Clayton Craig

Numerade Educator

Problem 26

Determine whether the series is convergent or divergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{n^4 + 1} $

Clayton Craig

Numerade Educator

Problem 27

Explain why the Integral Test can't be used to determine whether the series is convergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\cos \pi n}{\sqrt n} $

Clayton Craig

Numerade Educator

Problem 28

Explain why the Integral Test can't be used to determine whether the series is convergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\cos^2 n}{1 + n^2} $

Clayton Craig

Numerade Educator

Problem 29

Find the values of $ p $ for which the series is convergent.
$ \displaystyle \sum_{n = 2}^{\infty} \frac {{1}}{{n(\ln n)^P}} $

Clayton Craig

Numerade Educator

Problem 30

Find the values of $ p $ for which the series is convergent.
$ \displaystyle \sum_{n = 3}^{\infty} \frac {{1}}{{ n \ln n [\ln (\ln n)]^P}} $

JH

J Hardin

Numerade Educator

Problem 31

Find the values of $ p $ for which the series is convergent.
$ \displaystyle \sum_{n = 1}^{\infty} n(1 + n^2)^P $

JH

J Hardin

Numerade Educator

Problem 32

Find the values of $ p $ for which the series is convergent.
$ \displaystyle \sum_{n = 1}^{\infty} \frac {\ln n}{n^P} $

JH

J Hardin

Numerade Educator

Problem 33

The Riemann zeta-function $ \zeta $ is defined by
$ \zeta(x) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^x} $
and is used in number theory to study the distribution of prime numbers. What is the domain of $ \zeta $ ?

JH

J Hardin

Numerade Educator

Problem 34

Leonhard Euler was able to calculate the exact sum of the $ p- $ series with $ p = 2: $
$ \zeta (2) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^2} = \frac {\pi^2}{6} $
(See page 720.) Use this fact to find the sum of each series.
(a) $ \displaystyle \sum_{n = 2}^{\infty} \frac {1}{n^2} $
(b) $ \displaystyle \sum_{n = 3}^{\infty} \frac {1}{(n + 1)^2} $
(c) $ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{(2n)^2} $

Clayton Craig

Numerade Educator

Problem 35

Euler also found the sum of the $ p- $ series with $ p = 4: $
$ \zeta (4) = \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^4} = \frac {\pi^4}{90} $
Use Euler's result to find the sum of the series.
(a) $ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {3}{n} \right)^4 $
(b) $ \displaystyle \sum_{k = 5}^{\infty} \frac {1}{(k - 2)^4} $

JH

J Hardin

Numerade Educator

Problem 36

(a) Find the partial sum $ s_10 $ of the series $ \sum_{n = 1}^{\infty} 1/n^4. $ Estimate the error in using $ s_10 $ as an approximation to the sum of the series.
(b) Use (3) with $ n = 10 $ to give an improved estimate of the sum.
(c) Compare your estimate in part (b) with the exact value given in Exercise 35.
(d) Find a value of $ n $ so that $ s_n $ is within 0.00001 of the sum.

JH

J Hardin

Numerade Educator

Problem 37

(a) Use the sum of the first 10 terms to estimate the sum of the series $ \sum_{n = 1}^{\infty} 1/n^2. $ How good is this estimate?
(b) Improve this estimate using (3) with $ n = 10. $
(c) Compare your estimate in part (b) with the exact value given in Exercise 34.
(d) Find a value of $ n $ that will ensure that the error in the approximation $ s \approx s_n $ is less than 0.001.

JH

J Hardin

Numerade Educator

Problem 38

Find the sum of the series $ \sum_{n = 1}^{\infty} ne^{-2n} $ correct to four decimal places.

JH

J Hardin

Numerade Educator

Problem 39

Estimate $ \sum_{n = 1}^{\infty} (2n + 1)^{-6} $ correct to five decimal places.

JH

J Hardin

Numerade Educator

Problem 40

How many terms of the series $ \sum_{n = 2}^{\infty} 1/[n(\ln n)^2] $ would you need to add to find its sum to within 0.01?

JH

J Hardin

Numerade Educator

Problem 41

Show that if we want to approximate the sum of the series $ \sum_{n = 1}^{\infty} n^{-1.001} $ so that the error is less than 5 in the ninth decimal place, then we need to add more than $ 10^{11,301} $ terms!

JH

J Hardin

Numerade Educator

Problem 42

(a) Show that the series $ \sum_{n = 1}^{\infty} (\ln n)^2/n^2 $ is convergent.
(b) Find an upper bound for the error in the approximation $ s
\approx s_n. $
(c) What is the smallest value of $ n $ such that this upper bound is less that 0.05?
(d) Find $ s_n $ for this value of $ n. $

JH

J Hardin

Numerade Educator

Problem 43

(a) Use (4) to show that if $ s_n $ is the $ n $th partial sum of the harmonic series, then
$ s_n \le 1 + \ln n $
(b) The harmonic series diverges, but very slowly. Use part (a) to show that the sum of the first million terms is less than 15 and the sum of the first billion terms is less than 22.

JH

J Hardin

Numerade Educator

Problem 44

Use the following steps to show that the sequence
$ t_n = 1 + \frac {1}{2} + \frac {1}{3} + \cdot \cdot \cdot + \frac {1}{n} - \ln n $
has a limit. (The value of the limit is denoted by $ \gamma $ and is called Euler's constant.)
(a) Draw a picture like Figure 6 with $ f(x) = 1/x $ and interpret $ t_n $ as an area [or use (5)] to show that $ t_n > 0 $ for all $ n. $
(b) Interpret
$ t_n - t_{n + 1} = \left[ \ln \left(n + 1 \right) - \ln n \right] - \frac {1}{n + 1} $
as a difference of areas to show that $ \left\{ t_n \right\} $ is convergent.

JH

J Hardin

Numerade Educator

Problem 45

Find all positive values of $ b $ for which the series $ \sum_{n = 1}^{\infty} b^{\ln n} $ converges.

JH

J Hardin

Numerade Educator

Problem 46

Find all values of $ c $ for which the following series converges.
$ \displaystyle \sum_{n = 1}^{\infty} \left( \frac {c}{n} - \frac {1}{n + 1} \right) $

JH

J Hardin

Numerade Educator