Chapter 9.4, Problem 24E

Solution

i.

To state: The first six partial sums of the infinite series $-2+2-2+2-2+\cdots$ . 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 $-2+2-2+2-2+\cdots$ .

Explanation:

Consider the given infinite series:

  $-2+2-2+2-2+\cdots$

The first six partial sums are:

  $-2,0,-2,0,-2,0$

The partial sum appears to be oscillate and do not approach to a limit, which suggests that the series diverges.

Therefore, the infinite series is divergent.

ii.

To state: The first six partial sums of the infinite series $1-0.7-0.07-0.007-0.0007-\cdots$ . 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: $1-0.7-0.07-0.007-0.0007-\cdots$

Explanation:

Consider the given infinite series:

  $1-0.7-0.07-0.007-0.0007-\cdots$

The first six partial sums are:

  $1,0.3,0.23,0.223,0.2223,0.22223$

The partial sum appears to be approaching to a finite limit, which suggests that the series converges.

Therefore, the infinite series is convergent.