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We are asked to prove that the infinite series \[ \sum_{n=1}^{\infty }(-1)^{n+1}\frac{5}{7n+2} \] is conditionally convergent. ### Step 1: Determine if the series is alternating The given series is an alternating series because of the term \((-1)^{n+1}\). The general term is of the form: \[ a_n = (-1)^{n+1}\frac{5}{7n+2}. \] This is an alternating series, so we can apply the Alternating Series Test (Leibniz Criterion) to determine convergence. ### Step 2: Check the conditions of the Alternating Series Test The Alternating Series Test requires two conditions: 1. Monotonicity: The magnitude __________ ___ _______ ________ ________.
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