Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

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Short Answer

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?
\(\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} \)

Convergent.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definition

A convergent series is an infinite sum of a sequence of numbers\(\sum\limits_{n = 1}^\infty {{a_n}} \) that adds up to finite number.

A divergent series is a series that does not converge, so if add up all of the terms of the series\(\sum\limits_{n = 1}^\infty {{a_n}} \) don't have finite sum.

Find terms

Consider the series\(\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} \)

When\(n = 1\)

\(\begin{aligned}\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} &= \frac{1}{{\ln (1 + 1)}}\\ &= \frac{1}{{\ln (2)}}\\ &= 1.4427\end{aligned}\)

When\(n = 2\),

\(\begin{aligned}{s_2} &= \sum\limits_{n = 1}^2 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}}\\ &= 1.4427 + 0.91023\\ &= 2.35293\end{aligned}\)

find terms

When\(n = 3\),

\(\begin{aligned}{s_3} &= \sum\limits_{n = 1}^3 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}}\\ &= 3.0743\end{aligned}\)

When\(n = 4\),

\(\begin{aligned}{s_4} &= \sum\limits_{n = 1}^4 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}} + \frac{1}{{\ln (5)}}\\ &= 3.6956\end{aligned}\)

Find terms

When\(n = 5\),

\(\begin{aligned}{s_5} &= \sum\limits_{n = 1}^5 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}} + \frac{1}{{\ln (5)}} + \frac{1}{{\ln (6)}}\\ &= 4.2537\end{aligned}\)

When\(n = 6\),

\(\begin{aligned}{s_6} &= \sum\limits_{n = 1}^6 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}} + \frac{1}{{\ln (5)}} + \frac{1}{{\ln (6)}} + \frac{1}{{\ln (7)}}\\ &= 4.7676\end{aligned}\)

Find terms

When\(n = 7\),

\(\begin{aligned}{s_7} &= \sum\limits_{n = 1}^7 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}} + \frac{1}{{\ln (5)}} + \frac{1}{{\ln (6)}} + \frac{1}{{\ln (7)}} + \frac{1}{{\ln (8)}}\\ &= 5.2485{\rm{ }}\\{\rm{When }}n = 8\\{s_8} &= \sum\limits_{n = 1}^8 {\frac{1}{{\ln (n + 1)}}} \\ &= \frac{1}{{\ln (2)}} + \frac{1}{{\ln (3)}} + \frac{1}{{\ln (4)}} + \frac{1}{{\ln (5)}} + \frac{1}{{\ln (6)}} + \frac{1}{{\ln (7)}} + \frac{1}{{\ln (8)}} + \frac{1}{{\ln (9)}}\\ &= 5.7036\end{aligned}\)

Check convergence

Using result that if the sequence\(\left\{ {{s_n}} \right\}\) is convergent, then \(\mathop {\lim }\limits_{n \to \infty } {s_n} = s\)exists is a real number.

So, the series \(\sum {{a_n}} \)is called convergent and the sequence \(\left\{ {{s_n}} \right\}\)is divergent, then the series is called divergent.

Let\({a_n} = \frac{1}{{\ln (n + 1)}}\)then

\(\begin{aligned}\mathop {\lim }\limits_{n \to \infty } {a_n} &= \mathop {\lim }\limits_{n \to \infty } \frac{1}{{\ln (n + 1)}}\\ &= \frac{1}{{\ln (\infty )}}\\ &= \frac{1}{\infty }\\ &= 0{\rm{ (define) }}\end{aligned}\)

Therefore, the series is convergent.

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Essential Calculus: Early Transcendentals

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Short Answer

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?
\(\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} \)

Step by Step Solution

Definition

Find terms

find terms

Find terms

Find terms

Check convergence

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Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

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?\(\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} \)

Step by Step Solution

Definition

Find terms

find terms

Find terms

Find terms

Check convergence

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

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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?
\(\sum\limits_{n = 1}^\infty {\frac{1}{{\ln (n + 1)}}} \)