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.

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

Short answer

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

Step-by-step solution

Step 1. Given information

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

Step 2. The 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 term of the series ∑k=1∞ 1+ln kk are positive.

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

$\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k}$

Next find the $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}$for the given series.

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

Step 4. From the obtained values,

The value of $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=1$which is finite non zero number.

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

Therefore, $\sum _{k=1}^{\infty }{a}_k$is divergent.

Th given series is divergent.