Question

Describe how to apply the Alternating Series Test.

Short answer

Question: Determine if the following alternating series converges or diverges using the Alternating Series Test: \[ \sum_{n=1}^\infty (-1)^n \frac{n}{2n+1} \]

Step-by-step solution

Identify an alternating series

An alternating series is a series of the form: \[ \sum_{n=1}^\infty (-1)^{n-1}b_n \text{ or } \sum_{n=1}^\infty (-1)^n b_n, \] where \(b_n\) are non-negative terms. If the given series matches either of these forms, then it's an alternating series, and we can proceed with the Alternating Series Test.

Check the first condition - Decreasing terms

For the first condition, we need to check if the series has decreasing terms, i.e., \(b_{n+1} \leq b_n\) for all \(n\). In essence, the non-negative parts of the series should be decreasing or non-increasing.

Check the second condition - Limit of terms

For the second condition, we need to check if the limit of the terms, as \(n\) approaches infinity, is zero. That is, \[ \lim_{n\to\infty} b_n = 0. \] If this condition is met, then the terms are getting smaller and smaller as the series progresses.

Determine convergence or divergence

Once both conditions have been checked, if they both hold true, then the alternating series converges. If one or both of the conditions don't hold true, the Alternating Series Test is inconclusive, and another test may be needed to determine the convergence or divergence of the series. Note that the Alternating Series Test can only prove convergence, not divergence. If the test fails, it may still be possible to prove the divergence of the series using another test, such as the Comparison Test or the Ratio Test.