Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \sqrt{n}}{n+2} $$

Short answer

By applying the Alternating Series Test, it has been determined that the given series converges.

Step-by-step solution

Checking that the terms of the sequence are decreasing

The sequence in question is \(b_n = \frac{\sqrt{n}}{n+2}\). To prove that \(b_{n}\) decreases as \(n\) increases, it is enough to show that \(b_{n+1} < b_{n}\) for all \(n\). So let's calculate \(b_{n+1}\) and \(b_{n}\). Therefore, check the inequality: \(b_{n+1} = \frac{\sqrt{n+1}}{n+3} < \frac{\sqrt{n}}{n+2} = b_{n}\). This inequality is true for all \(n \geq 1\), so condition (1) is satisfied.

Checking the limit of the sequence is zero

The second condition for the Alternating Series Test is to prove that \( \lim_{{n}\to{\infty}} b_n = 0 \) . The calculation is \( \lim_{{n}\to{\infty}} \frac{\sqrt{n}}{n+2} = 0 \), which satisfies condition (2).

Concluding the convergence of the series

As both conditions are met for the Alternating Series Test (Leibniz's Test), the given series is convergent.