(a) What is a convergent sequence? (b) What is a convergent series? (c) What does lim n →∞ a n = 3 mean? (d) What does ∑ n = 1 ∞ a n = 3 mean?

Question

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Answer 1

Textbook Question

Chapter 11, Problem 1RCC

(a) What is a convergent sequence?
(b) What is a convergent series?
(c) What does lim_n_→∞ a_n = 3 mean?
(d) What does $∑ n = 1 ∞ a n = 3$ mean?

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

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Answer 2

Textbook Question

Chapter 11, Problem 1RCC

(a) What is a convergent sequence?
(b) What is a convergent series?
(c) What does lim_n_→∞ a_n = 3 mean?
(d) What does $∑ n = 1 ∞ a n = 3$ mean?

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

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Answer 3

Textbook Question

Chapter 11, Problem 1RCC

(a) What is a convergent sequence?
(b) What is a convergent series?
(c) What does lim_n_→∞ a_n = 3 mean?
(d) What does $∑ n = 1 ∞ a n = 3$ mean?

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

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Answer 4

Textbook Question

Chapter 11, Problem 1RCC

(a) What is a convergent sequence?
(b) What is a convergent series?
(c) What does lim_n_→∞ a_n = 3 mean?
(d) What does $∑ n = 1 ∞ a n = 3$ mean?

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

Answer 8

(a)

Expert Solution

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

To define: A convergent sequence.

Answer 13

Explanation of Solution

Definition:

If a sequence ${an}$ has a limit l, then the sequence is convergent sequence, which can be written as $limn→∞an=l$ . That is, $limn→∞an$ exists.

Examples:

The sequence ${1n}$ is a convergent sequence, which converges to 0.

Answer 14

(b)

Expert Solution

To determine

To define: Convergent series.

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

Answer 15

(b)

Expert Solution

Answer 16

(b)

Expert Solution

Answer 17

Expert Solution

Answer 18

To determine

To define: Convergent series.

Answer 19

Explanation of Solution

If the sequence of partial sums of the series is convergent, then the series is said to be convergent series.

Answer 20

(c)

Expert Solution

To determine

To describe: The meaning $limn→∞an=3$ .

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

Answer 21

(c)

Expert Solution

Answer 22

(c)

Expert Solution

Answer 23

Expert Solution

Answer 24

To determine

To describe: The meaning $limn→∞an=3$ .

Answer 25

Answer to Problem 1RCC

The sequence ${an}$ is closer to 3 as n approaches to larger number $(∞)$ .

Answer 26

Explanation of Solution

The sequence ${an}$ is converges to 3 as n tends to $∞$ . Here, the limit of the sequence ${an}$ is 3.

Answer 27

(d)

Expert Solution

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer to Problem 1RCC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

Answer 28

(d)

Expert Solution

Answer 29

(d)

Expert Solution

Answer 30

Expert Solution

Answer 31

To determine

To explain: The meaning of $∑n=1∞an=3$ .

Answer 32

Answer to Problem 1RCC

The sum of the series is 3.

Answer 33

Explanation of Solution

The sequence of partial sums ${an}$ is closer to 3 as n approaches to larger number.

That is, the sequence of partial sums converges to 3 as n tends to $∞$ .

Therefore, the sum of the series is 3.

Answer 34

Ch. 11.1 - (a) What is a sequence? (b) What does it mean to...Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E Ch. 11.1 - Prob. 4E Ch. 11.1 - Prob. 5E Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - Prob. 8E Ch. 11.1 - Prob. 9E Ch. 11.1 - Prob. 10E

(a) What is a convergent sequence? (b) What is a convergent series? (c) What does lim n →∞ a n = 3 mean? (d) What does ∑ n = 1 ∞ a n = 3 mean?

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Explanation of Solution

Explanation of Solution

Answer to Problem 1RCC

Explanation of Solution

Answer to Problem 1RCC

Explanation of Solution

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Chapter 11 Solutions