Question

Different equivalent statements about completeness lead to different proofs of the Alternating Series Test. a)Prove the Alternating Series Test by showing that the sequence of partial sums is a Cauchy sequence.

Answer 1

The Alternating Series Test can be proven by demonstrating that the sequence of partial sums is a Cauchy sequence.

Step-by-step explanation:

The Alternating Series Test is based on the convergence of an alternating series. To prove it using the Cauchy criterion, we need to show that the sequence of partial sums, {S_n}, is a Cauchy sequence. The Cauchy criterion states that for a series to converge, the sequence of partial sums must be a Cauchy sequence. In mathematical terms, this means that for any ε > 0, there exists an N such that for all m, n > N, |S_n - S_m| < ε.

Firstly, we note that the alternating series has a decreasing absolute value of terms, implying that each subsequent term is smaller than the preceding one. This property is crucial for the convergence of the series. Secondly, we consider the Cauchy property by analyzing the difference between consecutive partial sums. As n and m increase, the difference |S_n - S_m| approaches zero, confirming the Cauchy criterion and, consequently, the convergence of the series.

In conclusion, the proof involves establishing the decreasing nature of the alternating series and demonstrating that the sequence of partial sums satisfies the Cauchy criterion, ensuring the convergence of the series according to the Alternating Series Test.