Chapter 10.5, Problem 34E

(a)

To determine

To find: The series that is a divergent series.

Answer to Problem 34E

Since given series is an alternating series. So first make partial sum larger than one. Then add negative term to make it less than $-2$ . Continue this manner until the sequence of partial sum s diverges that means it is the divergent series.

Explanation of Solution

Given information: ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{n\ln n}}$

Calculation: The given series is an alternating series. If its index $n$ is odd, then term will be positive and if it is even, then the term will be negative.

Keep adding positive terms until $s_n=1$ . Add negative term until $s_n=-2$ . Repeat this procedure in this manner until the partial sum oscillates between increasing values in both directions and thus diverges.

Therefore, we can say that series is divergent in nature because it does not converge.

(b)

To determine

To find: The series that is a convergent to 4.

Answer to Problem 34E

First add positive terms! If the partial sum is larger than 4, add negative terms, and if the partial sum is less than 4, add positive terms, so that the partial sums are ever growing closer to 4. Then it is a convergent to 4.

Explanation of Solution

Given information: ${\displaystyle \sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}\frac{1}{n\ln n}}$

Calculation: Since terms of the series is alternate between positive and negative.

Hence, make the partial sum is larger than four by adding positive terms. Then, make the partial sum is lower than four by adding negative terms. Repeat this process indefinitely. At the end $s_n$ is getting closer and closer to four. Hence, the rearranged series must be convergent to four.