Chapter 9.1, Problem 35E

a.

To determine

To find what happening to the sequence of partial sums.

Answer to Problem 35E

The partial sum tends towards infinity.

Explanation of Solution

Given:

The given series $S={\displaystyle \sum _{n=0}^{\infty }2n}$ .

Formula used:

If the sequence of partial sum has a limit as $n\to \infty$ , the series converges otherwise the sequence is divergence. For series to be convergence the term should approach to zero. That is common ratio should be less than $1$ . That is $\left|r\right|<1$ , where $r$ is the common ratio.

Calculation:

Given that the geometric series converges.

  $S={\displaystyle \sum _{n=0}^{\infty }2n}$

Expanding the series

  $S=2+4+6+8+10+\mathrm{......}$

The terms are positive and do not approach to zero, the partial sum tends towards infinity

b.

To determine

To find what happening to the sequence of partial sums.

Answer to Problem 35E

The partial sum of the series are alternately $0$ and $1$ .

Explanation of Solution

Given:

The given series $S={\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}$ .

Formula used:

If the sequence of partial sum has a limit as $n\to \infty$ , the series converges otherwise the sequence is divergence. For series to be convergence the term should approach to zero. That is common ratio should be less than $1$ . That is $\left|r\right|<1$ , where $r$ is the common ratio.

Calculation:

Given that the geometric series converges.

  $S={\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}$

Expanding the series

  $S=1-1+1-1+1-1+\mathrm{..........}$

Taking terms in pair of terms

  $S=\left(1-1\right)+\left(1-1\right)+\left(1-1\right)+\mathrm{..........}$

Depending on the number of terms even or odd, partial sum of the series are alternately $0$ and $1$ .

c.

To determine

To find what happening to the sequence of partial sums.

Answer to Problem 35E

The magnitude towards infinity.

Explanation of Solution

Given:

The given series $S={\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}(2n)$ .

Formula used:

If the sequence of partial sum has a limit as $n\to \infty$ , the series converges otherwise the sequence is divergence. For series to be convergence the term should approach to zero. That is common ratio should be less than $1$ . That is $\left|r\right|<1$ , where $r$ is the common ratio.

Calculation:

Given that the geometric series converges.

  $S={\displaystyle \sum _{n=0}^{\infty }{(-1)}^n}(2n)$

Expanding the series

  $S=-2+4-6+8-10+\mathrm{..........}$

The partial sums alternate between positive and negative while magnitude increases towards infinity.