In the present paper, we treat an infinite series ($alpha$, $eta$)-metric L =$eta$$^2$/($eta$-$alpha$). First, we find the conditions that a Finsler metric F$^{n}$ with the metric above be a Berwald space, a Douglas space, and a projectively flat Finsler space, respectively. Next, we investigate the condition that a two-dimensional Finsler space with the metric above be a Landsbeg space. Then the differential equations of the geodesics are also discussed.??景??楯渠慮????楣?楯湳?????k楣?h???????歲?