Question

In Exercises \(1-16,\) determine the convergence or divergence of the series. \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\)

Short answer

The series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) converges.

Step-by-step solution

Identify the series type

The series \(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}\) is an alternating series because its terms alternate in sign.

Apply first condition of the Alternating Series Test

The first condition of the Alternating Series Test is that the limit as n approaches infinity of the absolute value of the n-th term should be zero. The absolute value of the n-th term here would be \(\frac{1}{n}\). As n approaches infinity, \(\frac{1}{n}\) approaches 0.

Apply second condition of the Alternating Series Test

The second condition of the test is that the absolute values of the terms should be decreasing. In this series, the absolute values of the terms are \(\frac{1}{n}\), which is decreasing as n increases.

Conclude the result

Since the two conditions of the Alternating Series Test are both satisfied, the given series converges.