Answer
$\ln (\dfrac{5}{3}) \approx 0.510825624$
Work Step by Step
Since, we have the form of the given series as: $\ln (1+x)=x-\dfrac{x^2}{2}+\dfrac{x^3}{3}-....+(-1)^{n-1}\dfrac{x^n}{n}$
Here, in the given problem $x=\dfrac{2}{3}$
Thus, the sum of the series is:
$\ln (1+\dfrac{2}{3})=\ln (\dfrac{3+2}{3})=\ln (\dfrac{5}{3}) \approx 0.510825624$
