91Ó°ÊÓ

Absolute convergence means that the series obtained by taking the absolute values of the terms converges. In other words, if we have a series ∑a_n, then it is absolutely convergent if ∑|a_n| converges. Absolute convergence is stronger than convergence, and it implies convergence.

A sequence of partial sums {S_n} converges if and only if for any given ε > 0, there exists a positive integer N such that for all natural numbers m, n with m > n ≥ N, we have |S_m - S_n| < ε. This is called the Cauchy criterion for convergence. We will use this criterion to show why absolute convergence implies convergence.

Let's consider an absolutely convergent series ∑a_n. This means ∑|a_n| converges. To prove convergence of ∑a_n, we need to show that the sequence of partial sums {S_n} satisfies the Cauchy criterion for convergence. Suppose ε > 0 is given. Since ∑|a_n| converges, there exists a positive integer N such that for all natural numbers m, n with m > n ≥ N, we have: |(\sum_{k=n+1}^{m} |a_k|)| < ε. Now, let's consider the sequence of partial sums {S_n} for the original series ∑a_n. We have: |S_m - S_n| = |(\sum_{k=n+1}^{m} a_k)| ≤ (\sum_{k=n+1}^{m} |a_k|). The inequality is justified by the triangle inequality. Since we already have |(\sum_{k=n+1}^{m} |a_k|)| < ε for m > n ≥ N, this means: |S_m - S_n| ≤ (\sum_{k=n+1}^{m} |a_k|) < ε. Thus, the sequence of partial sums {S_n} satisfies the Cauchy criterion for convergence, and it follows that the original series ∑a_n converges. This proves that absolute convergence implies convergence.