Question

Determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty} \frac{1}{n} \cos n \pi $$

Short answer

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

Step-by-step solution

Identify the type of the series

Establish that this is an alternating series. An alternating series is defined as one that alternates between positive and negative terms. This is true for \( \sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi \) because \(\cos n\pi\) alternates between 1 and -1 as n varies between even and odd numbers.

Applying the Alternating Series Test

The Alternating Series Test states that the alternating series \(\sum_{n=1}^{\infty} (-1)^n a_n\), where each \(a_n > 0\), converges if it satisfies two conditions: (1) The terms \(a_n\) are decreasing and (2) The limit as n approaches infinity of \(a_n\) equals zero. Here \(a_n = \frac{1}{n}\). The terms \(a_n = \frac{1}{n}\) are clearly decreasing and its limit as n approaches infinity is indeed zero.

Conclude

Based on the Alternating Series Test, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \cos n\pi \) converges since both conditions of the test are satisfied. It is important to remember this conclusion relies on the series being an alternating series and meeting all the conditions of the Alternating Series Test.