Final answer:
The alternating series test states that if a series satisfies certain conditions, including the Cauchy criterion, it is convergent. To prove this, we consider the difference between two partial sums and show that it can be made arbitrarily small. Therefore, the sequence of partial sums is Cauchy and the alternating series converges.
Step-by-step explanation:
The alternating series test states that if a series, ∑⟨subscript⟩n=1⟨/subscript⟩∞(−1)^⟨subscript⟩n+1⟨/subscript⟩⟨subscript⟩n⟨/subscript⟩, satisfies two conditions, then it is convergent. One of these conditions is that the sequence of partial sums of the series, {S⟨subscript⟩n⟨/subscript⟩}, is Cauchy. To prove this, consider the difference between two partial sums, |S⟨subscript⟩n⟨/subscript⟩ - S⟨subscript⟩m⟨/subscript⟩|. By applying the triangle inequality and rearranging the terms, it can be shown that for any ε > 0, there exists an integer N such that for all n, m > N, |S⟨subscript⟩n⟨/subscript⟩ - S⟨subscript⟩m⟨/subscript⟩| < ε. Therefore, the sequence of partial sums is Cauchy and the alternating series converges.