   Chapter 11, Problem 1RCC

Chapter
Section
Textbook Problem

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

(a)

To determine

To define: A convergent sequence.

Explanation

Definition:

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

Examples:

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

(b)

To determine

To define: Convergent series.

Explanation

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

(c)

To determine

To describe: The meaning limnan=3.

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

Explanation

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

(d)

To determine

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

The sum of the series is 3.

Explanation

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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