sum_(n=1)^infty ((-1)^n-1(2 n+1))/(n(n+1))=1 old góster

The given series is $\sum _{n=1}^{\infty }(-1)^{n-1}(2n+1)$. We need to determine if this series converges to 1.<br /><br />To do this, we can use the Alternating Series Test. The Alternating Series Test states that if the terms of an alternating series decrease monotonically and approach zero, then the series converges.<br /><br />Let's apply the Alternating Series Test to our series:<br /><br />1. The terms of the series are alternating in sign, since we have $(-1)^{n-1}$.<br /><br />2. The terms of the series are decreasing monotonically, since $(2n+1)$ is increasing.<br /><br />3. The terms of the series approach zero as $n$ approaches infinity.<br /><br />Since all three conditions of the Alternating Series Test are satisfied, we can conclude that the series converges.<br /><br />To find the sum of the series, we can use the formula for the sum of an alternating series:<br /><br />$\sum _{n=1}^{\infty }(-1)^{n-1}(2n+1) = 1$<br /><br />Therefore, the correct answer is 1.