Chapter 8, Problem 6RCC

(a)

To determine

To find: The absolute convergent series.

Answer to Problem 6RCC

Theabsolute convergent series is defined.

Explanation of Solution

Calculation:

The absolute convergent series is the series of the form ${\displaystyle \sum a_n}$ only when the corresponding series is of absolute values of ${\displaystyle \sum \left|a_n\right|}$ is convergent.

(b)

To determine

To find: The statement about the convergent series.

Answer to Problem 6RCC

The statement is that if the series is ${\displaystyle \sum \left|a_n\right|}$ convergent, then ${\displaystyle \sum a_n}$ converges absolutely.

Explanation of Solution

Calculation:

Consider the series is ${\displaystyle \sum a_n}=1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+\mathrm{...}$ absolutely convergent and as the series is absolute value is ${\displaystyle \sum a_n}=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\mathrm{...}$ that is being the geometric sequence with the ratio $\left(\frac{1}{2}<1\right)$ is also convergent.

Thus, the statement is that if the series is ${\displaystyle \sum \left|a_n\right|}$ convergent, then ${\displaystyle \sum a_n}$ converges absolutely.