Question

Determine whether the series converges conditionally or absolutely, or diverges. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}} $$

Short answer

The given series converges conditionally, not absolutely.

Step-by-step solution

Absolute Convergence Test

To perform the absolute convergence test we calculate the sum of the absolute values of the terms: \( \sum_{n=1}^{\infty} \left| \frac{(-1)^{n+1}}{\sqrt{n}} \right| = \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \). The series \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) is a p-Series with p = 0.5, and we know that a p-Series converges if p > 1 and diverges if p ≤ 1. So, this series diverges.

Alternating Series Test

Since the absolute version of the series diverged, it cannot be absolutely convergent. However, it could be conditionally convergent. This is where the alternating series test comes into play. The alternating series test states that the alternating series converges if its sequence of terms decreases towards 0. Here, the series is \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{\sqrt{n}}\) The terms \(\frac{1}{\sqrt{n}}\) are decreasing and their limit as n goes to infinity is 0. Therefore, the series is conditionally convergent under the Alternating Series Test.