The given series is:\[\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{5 n+7}\]To check whether the given series converges or diverges, let's first analyze the series.The given series is an alternating series, which means the series is of the form \[\sum_{n=1}^{\infty}(-1)^{n-1} b_n,\]where the given series can be represented as $b_n=\frac{n}{5n+7}$.Let's evaluate the limit of $b_n$ as $n$ approaches infinity, which can be done by applying the limit test as shown below:\[\lim_{n \rightarrow \infty} \frac{n}{5n+7} = \lim_{n \rightarrow \infty} \frac{1}{5+\frac{7}{n}} = \frac{1}{5}\]Since $b_n$ is positive and the limit is not equal to zero, we can say that the series diverges by the Alternating Series Test. Therefore, the given series is divergent. Main answer:The given series is divergent.Explanation:We can conclude that the series diverges by the Alternating Series Test as $b_n=\frac{n}{5n+7}$ is positive and the limit is not equal to zero, so the series is divergent.Conclusion:Thus, the given series \[\sum_{n=1}^{\infty} \frac{(-1)^{n+1} n}{5 n+7}\]converges to a value.