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.

$\sum _{k=0}^{\infty }\frac{{(-2)}^k}{1+2^k}$

Short answer

The series $\sum _{k=0}^{\infty }\frac{{(-2)}^k}{1+2^k}$ diverges..

Step-by-step solution

Step 1. Given Information.

The series:

$\sum _{k=0}^{\infty }\frac{{(-2)}^k}{1+2^k}$

Step 2. Find limk→∞ak.

$$\begin{gathered} \underset{k\to \infty }{lim}{a}_k=\underset{k\to \infty }{lim}\left(\frac{{(-2)}^k}{1+2^k}\right) \\ \ne 0 \end{gathered}$$

Since the limit of the sequence does not converges to zero, so the sequence converge.

The sequence has to be either converge or diverge.

So the sequence diverges.