Chapter 11, Problem 1CC

(a) What is a convergent sequence?

(b) What is a convergent series?

(c) What does lim_n→∞a_n= 3 mean?

(d) What does ${\displaystyle {\sum }_{n=1}^{\infty }{a}_n}=3$ mean?

To determine

To define: A convergent sequence.

Explanation of Solution

Definition:

If a sequence $\left\{a_n\right\}$ has a limit l, then the sequence is convergent sequence, which can be written as $\underset{n\to \infty }{\lim }{a}_n=l$. That is, $\underset{n\to \infty }{\lim }{a}_n$ exists.

Examples:

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

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.

To determine

To describe: The meaning $\underset{n\to \infty }{\lim }{a}_n=3$.

Answer to Problem 1CC

The sequence $\left\{a_n\right\}$ is closer to 3 as n approaches to larger number $\left(\infty \right)$.

Explanation of Solution

The sequence $\left\{a_n\right\}$ is converges to 3 as n tends to $\infty$. Here, the limit of the sequence $\left\{a_n\right\}$ is 3.

To determine

To explain: The meaning of ${\displaystyle \sum _{n=1}^{\infty }{a}_n}=3$.

Answer to Problem 1CC

The sum of the series is 3.

Explanation of Solution

The sequence of partial sums $\left\{a_n\right\}$ is closer to 3 as n approaches to larger number.

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

Therefore, the sum of the series is 3.