Question

Use any convergence test from this section or the previous section to determine whether the series in Exercises $31-48$converge or diverge. Explain how the series meets the hypotheses of the test you select.

$\sum _{k=1}^{\infty }{k}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$.

Short answer

The series$\sum _{k=1}^{\infty }{k}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$is divergent.

Step-by-step solution

Step 1. Given information

$\sum _{k=1}^{\infty }{k}^{\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$.

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

1. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=L$, where $L$is any positive real number.
2. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=0$and role="math" localid="1649222738593" $\sum _{k=1}^{\infty }{b}_k$converges, then $\sum _{k=1}^{\infty }{a}_k$also converges.
3. If $\underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\infty$, and $\sum _{k=1}^{\infty }{b}_k$diverges, then$\sum _{k=1}^{\infty }{a}_k$also diverges.

Step 3. The terms of the series ∑k=1∞ 1k12 are positive.

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

$\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

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

$$\begin{gathered} \underset{k\to \infty }{\lim }\frac{a_k}{b_k}=\underset{k\to \infty }{\lim }\frac{{\displaystyle \frac{1}{k^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}}{{\displaystyle \frac{1}{k^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}}} \\ =\underset{k\to \infty }{\lim }1 \\ =1 \end{gathered}$$

Step 4. From the obtained values,

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

The series $\sum _{k=1}^{\infty }{b}_k=\sum _{k=1}^{\infty }\frac{1}{k^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$is divergent by $p-$series test.

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

Hence, the given series is divergent.