the mathematical limit as approaches 1 is well defined and the corresponding diversity is calculated with the following equation: which is the exponential of the Shannon entropy calculated with natural logarithms (see above). In other domains, this statistic is also known as the perplexity. The general equation of diversity is often written in the form and the term inside the parentheses is called the basic sum. Some popular diversity indices correspond to the basic sum as calculated with different values of . Sensitivity of the diversity value to rare vs. abundant species The value of is often referred to as the order of the diversity. It defines the sensitivity of the true diversity to rare vs. abundant species by modifying how the weighted mean of the species' proportional abundances is calculated. With some values of the parameter , the value of the generalized mean assumes familiar kinds of weighted means as special cases. In particular, corresponds to the weighted harmonic mean, to the weighted geometric mean, and to the weighted arithmetic mean. As approaches infinity, the weighted generalized mean with exponent approaches the maximum value, which is the proportional abundance of the most abundant species in the dataset. Generally, increasing the value of increases the effective weight given to the most abundant species. This leads to obtaining a larger value and a smaller true diversity () value with increasing . When , the weighted geometric mean of the values is used, and each species is exactly weighted by its proportional abundance (in the weighted geometric mean, the weights are the exponents). When , the weight given to abundant species is exaggerated, and when , the weight given to rare species is. At , the species weights exactly cancel out the species proportional abundances, such that the weighted mean of the values equals even when all species are not equally abundant. At , the effective number of species, , hence equals the actual number of species . In the context of diversity, is generally limited to non-negative values. This is because negative values of would give rare species so much more weight than abundant ones that would exceed . Richness Richness simply quantifies how many different types the dataset of interest contains. For example, species richness (usually noted ) of a dataset is the number of species in the corresponding species list. Richness is a simple measure, so it has been