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Let f(n)= Sum binomial(n,k)^(-1). First, we show that f:N to Q_p is nowherecontinuous in the p-adic topology. If x is a p-adic integer, we say that f(x)is p-definable if lim f(x_j) exists in Q_p, where x_j denotes the jth partialsum for x. We prove that f(-1) is p-definable for all primes p, and if p isodd, then -1 is the only element of Z_p - N for which f(x) is p-definable. Forp=2, we show that if k is a positive integer, then f(-k-1) is not 2-definable,but that if the 1's in the binary expansion of x are eventually very sparse,then f(x) is 2-definable.  Some of our proofs require that p satisfy one of two conditions. There arethree small primes that do not satisfy the relevant condition, but our theoremscan be proved directly for these primes. No other prime less than 100,000,000fails to satisfy the conditions.

For which p-adic integers x can Sum_k binomial(x,k)^(-1) be defined?