Question

In Exercises $21-30$use one of the comparison tests to determine whether the series converges or diverges. Explain how the given series satisfies the hypotheses of the test you use.

role="math" localid="1649219965557" $\sum _{k=1}^{\infty }\frac{1+\ln k}{k^3}$.

Short answer

The series$\sum _{k=1}^{\infty }\frac{1+\ln k}{k^3}$is convergent.

Step-by-step solution

Step 1. Given information

$\sum _{k=1}^{\infty }\frac{1+\ln k}{k^3}$.

Step 2. The limit comparison test states that for ∑k=1∞ ak and ∑k=1∞ bk be two series with positive terms then,

1. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, where $L$is any positive real number, then either both converge or both diverge.
2. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$, and $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }{a}_k$converges.
3. If$\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$, and$\sum _{k=1}^{\infty }{b}_k$diverges, then$\sum _{k=1}^{\infty }{a}_k$diverges.

Step 3. The terms of the series ∑k=1∞ 1+ln kk3 are positive.

The expression $1+\ln k$satisfies the following inequality:

$1+\ln k\le k$.

Find $\sum _{k=1}^{\infty }{b}_k$for the given series.

$$\begin{gathered} \sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{k}{k^3} \\ =\sum _{k=1}^{\infty }\frac{1}{k^2} \end{gathered}$$

Step 4. Next find limk→∞ akbk for the given series.

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\underset{k\to \infty }{\lim }\frac{{\displaystyle \frac{1+\ln k}{k^3}}}{{\displaystyle \frac{1}{k^2}}} \\ =\underset{k\to \infty }{\lim }\frac{1+\ln k}{k} \end{gathered}$$

$=0$[ using L'Hopital's rule]

Step 5. From the obtained values,

The value of $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$which is a finite number.

The value of $\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k^2}$is convergent by $p-$series test.

Therefore, the series $\sum _{k=1}^{\infty }{a}_k$is also convergent.

Hence, the given series is convergent.