Chapter 10.1, Problem 35E

(a)

To determine

To find: The sequence of partial sums of ${\displaystyle \sum _{n=1}^{\infty }2n}$

Answer to Problem 35E

The terms are positive and do not approach to zero, the partial sum tend toward infinity

Explanation of Solution

Given information:

The given series is shown below,

  ${\displaystyle \sum _{n=1}^{\infty }2n}$

Formula used:

  $S=2+4+6+8+10+\mathrm{....}$

Calculation:

  $S={\displaystyle \sum _{n=1}^{\infty }2n}$

On expanding the series

  $S=2+4+6+8+10+\mathrm{....}$

Since the terms are positive and do not approach to zero, the partial sum tend toward infinity.

Conclusion:

The terms are positive and do not approach to zero, the partial sum tend toward infinity

(b)

To determine

To find: The sequence of partial sums of ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}$

Answer to Problem 35E

The number of terms even or odd, partial sum of the series is alternately 0 and 1.

Explanation of Solution

Given information:

The given series is shown below,

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

Formula used:

  $S={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}=1-1+1-1+1-1+\mathrm{.....}$

Calculation:

  $S={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}$ On expanding the series

  $S={\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}=1-1+1-1+1-1+\mathrm{.....}\text{Taking pair of terms}$

  $S=\left(1-1\right)+\left(1-1\right)+\left(1-1\right)+\left(1-1\right)+\mathrm{........}$ Depending on the number of terms even or odd, partial sum of the series are alternately 0 and 1.

Conclusion:

The number of terms even or odd, partial sum of the series is alternately 0 and 1.

(c)

To determine

The sequence of partial sums of ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n}\left(2n\right)$

Answer to Problem 35E

The partial sums alternate between positive and negative while their magnitude increases toward infinity

Explanation of Solution

Given information:

The given series is shown below,

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

Formula used:

  $S={\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n}\left(2n\right)=-2+4-6+8-10+\mathrm{.....}$

Calculation:

  $S={\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n}\left(2n\right)$ On expanding the series

  $S={\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^n}\left(2n\right)=-2+4-6+8-10+\mathrm{.....}$ Therefore, the partial sums alternate between positive and negative while their magnitude increases toward infinity.

Conclusion:

The partial sums alternate between positive and negative while their magnitude increases toward infinity