Question

Conditional and absolute convergence: For each of the series that follow, determine whether the series converges absolutely, converges conditionally, or diverges. Explain the criteria you are using and why your conclusion is valid.

$\sum _{k=1}^{\infty }{(-1)}^{k+1}\sin \left(\frac{\mathrm{\pi }}{k}\right)$

Short answer

The series $\sum _{k=1}^{\infty }{(-1)}^{k+1}\sin \left(\frac{\mathrm{\pi }}{k}\right)$converges conditionally.

Step-by-step solution

Step 1. Given Information.

The series:

$\sum _{k=1}^{\infty }{(-1)}^{k+1}\sin \left(\frac{\mathrm{\pi }}{k}\right)$

Step 2. By Alternating Series Test.

According to the Alternating Series Test, the sequence $a_{k+1}<a_k$for every . Then the alternating series $a_{k+1},a_k$both converges.

Step 3. Find ak+1.

$$\begin{gathered} a_k=\sin \left(\frac{\mathrm{\pi }}{k}\right) \\ {a}_{k+1}=\sin \left(\frac{\mathrm{\pi }}{k+1}\right) \\ {a}_{k+1}<a_k \end{gathered}$$

So the series is monotonic decreasing sequence.

Step 4. Find limk→∞ak.

$$\begin{gathered} \underset{k\to \infty }{lim}{a}_k=\underset{k\to \infty }{lim}\sin \left(\frac{\mathrm{\pi }}{k}\right) \\ =0 \end{gathered}$$

So the series converges.

Step 5. Find bk.

To find if the series converges absolutely or conditionally $b_k=\sin \frac{\mathrm{\pi }}{k}$which is a dominant term.

$\underset{k\to \infty }{lim}{b}_k$is divergent by p-series test.

So the given series is conditionally convergent.