Question

Determine whether the series in Exercises 21–23 converge or diverge. Note that these series do not satisfy the criteria of the alternating series test.

$$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}+...$$

Short answer

The given series diverges.

Step-by-step solution

Step 1. Given Information

• Given series, $$1-\frac{1}{2}-\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}-\frac{1}{4}+...$$
• These series do not satisfy the criteria of the alternating series test.

Step 2. Explanation

Here, the given series can be defined as, $$u_{n}=(-1)^{n+1}$$

And, it has two limit points.

$$\implies$$ The given series does not converge.

Therefore, the given series diverges.