what is 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 +1/9 - ...? I know that it is converging because it is alternating series with terms getting smaller to zero. but I do not know what it converges to value

$ S = 1 - \frac{1}{2} + \frac{1}{3} - \dots - \frac{1}{2n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2n} - 2 ( \frac{1}{2} + \frac{1}{4} + \dots + \frac{1}{2n}) =  1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{2n} - 2 \frac{1}{2} ( 1 + \frac{1}{2} + \frac{1}{3} + ... \frac{1}{n} ) $
Call $ H_n = 1 + \frac{1}{2} + ... \frac{1}{n} $ the harmonic series 
We know that $ H_n - \ln n\,$ converges to the Euler-Mascheroni constant
$\gamma = 0.57...$
$S = H_{2n} - H_n = H_{2n} - \ln{2n} +  H_n + \ln n + \ln 2n - \ln n = (H_{2n} - \ln 2n ) - ( H_n - \ln n ) + ln ( \frac{2n}{n} ) \to \ln( \frac{2n}{n} ) = \ln 2$
