Chapter 9.4, Problem 23E

Solution

i.

To state: The first six partial sums of the infinite series $0.3+0.03+0.003+0.0003+\cdots$ . If the sum has a finite limit, write “convergent”. If not, write “divergent”.

The infinite series is convergent.

Given information:

The given infinite series is $0.3+0.03+0.003+0.0003+\cdots$ .

Explanation:

Consider the given infinite series:

  $0.3+0.03+0.003+0.0003+\cdots$

The first six partial sums are:

  $0.3,0.33,0.333,0.3333,0.33333,0.333333$

The partial sum appears to be approaching a limit of $0.\overline{3}$ , which suggests that the series converges.

Therefore, the infinite series is convergent.

ii.

To state: The first six partial sums of the infinite series $1-2+3-4+5-6+\cdots$ . If the sum has a finite limit, write “convergent”. If not, write “divergent”.

The infinite series is divergent.

Given information:

The given infinite series is: $1-2+3-4+5-6+\cdots$

Explanation:

Consider the given infinite series:

  $1-2+3-4+5-6+\cdots$

The first six partial sums are:

  $1,-1,2,-2,3,-3$

The partial sum appears to be approaching to no limit, which suggests that the series diverges.

Therefore, the infinite series is divergent.