Question

Define a series that satisfies the Cauchy criteria.

A. A series with decreasing terms.
B. A series with bounded terms.
C. A series where the sum of differences of consecutive terms approaches zero.
D. A series with alternating terms.

Answer 1

Option C is the correct definition of a series satisfying the Cauchy criteria, which means the sum of the differences of consecutive terms approaches zero, indicating convergence.

Step-by-step explanation:

A series satisfies the Cauchy criteria if for any given positive number ε, there exists a number N such that for all n ≥ N, the absolute value of the sum of the series' terms from nth term to any mth term (with m > n) is less than ε. This condition implies that as more terms are added, the increments to the sum get smaller and smaller, eventually becoming arbitrarily small, indicating that the series is converging to a certain value.

Option C from the given choices most closely aligns with the Cauchy criteria as it states that the series is one where the sum of differences of consecutive terms approaches zero. This reflects the idea that the partial sums of the series become stable as the series progresses, which is a requirement for convergence based on the Cauchy criteria.