Is there an evolutionary purpose to armpit perspiration?
There may be an evolutionary purpose to underarm perspiration, as alluded to in the "hidden hint" link of @DevashishDas, and also hinted at by the answer by @Chris. The apocrine sweat glands of our armpits produce steroids. One group of steroids secreted by the axillary apocrine glands are sex pheromones.  Although we may not consciously notice the chemical odors of sex pheromones, they do likely affect our behaviors.

For example, Wedekind et al. showed that females showed distinct preferences for certain male odors. Women tended to prefer the odors of males that were genetically different at the major histocompatibility complex (MHC) loci, a group of genes associated with our immune response system. Greater genetic variation of MHC genes generally translates to greater ability to combat disease.  Females that select males that are genetically different at the MHC would produce offspring with greater genetic variability at the complex, and therefore more likely to have greater resistance to disease or ability to fight disease. 

These results suggest that the pheromones secreted in underarm perspiration is a sexually selected trait in humans. If so, underarm perspiration would have an evolutionary (fitness) advantage.