We present analysis of data obtained from the Medium Energy proportional counter array (ME) from the EXOSAT observations of the X-ray burst source Ser X-1 made in 1983 and 1985. From the combined analysis of the spectral and fast variability properties, we show that the source is an atoll source in the banana state during the observations. The persistent spectra of the 1985 data, for which reliable background data are available, are described well by the two-component model composed of a power-law with an exponential cutoff and a blackbody. The spectral study of two bursts, which were observed during the 1985 observation, shows no clear evidence for a departure from blackbody. One of the bursts shows a weak secondary maximum, which is more prominent in the low energy band than in the high energy band, and the burst shows possible evidence of an absorption line feature at about 4.1 keV.