Geometric distribution with p = 1

I found in the standard that that std::geometric distribution requires 0 < p < 1 for its termination probability and that std::negative_binomial_distribution requires 0 < p ≤ 1. I'm not sure why the former should be misbehaving for p = 1 (it should just always return 0 with unit probability), especially in the light of the claim that

std::geometric_distribution<>(p) is exactly equivalent to std::negative_binomial_distribution<>(1, p).

Only that I did not find the latter explicitly said in the standard draft. Should I, just to be careful, always use std::negative_binomial_distribution instead of std::geometric distribution when the parameter p can (beyond my control) go as high as 1?

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Interestingly, the various sources seem to be quite sloppy on the requirements. For example, www.cplusplus.com and the Boost documentation both say that for std::geometric distribution 0 ≤ p ≤ 1, while p = 0 makes no sense mathematically. (Boost also says that geometric distro is equivalent with negative binomial with r = 1, cplusplus does not.)