In summary, alternating binomial sums are mathematical expressions involving the sum of alternating binomial coefficients, which are numerical values in the expansion of a binomial expression. The formula for these sums is (-1)^k * nCk * x^(n-k) * y^k, with various real-life applications in probability, statistics, and physics. They are significant in mathematics as they demonstrate the relationship between binomial coefficients and expansions, and have patterns and properties such as the sum always equaling zero and following specific patterns.
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Show that for all positive integers ##n##, $$\binom{n}{1} - \frac{1}{2}\binom{n}{2} + \cdots + (-1)^{n-1}\frac{1}{n}\binom{n}{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$$