Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

Verified step by step guidance

1. Recall that the series \( \sum a_k \) converges and each term \( a_k > 0 \). Since the terms are positive and the series converges, it follows that \( a_k \to 0 \) as \( k \to \infty \).
2. Because \( a_k \to 0 \), there exists an index \( N \) such that for all \( k > N \), \( a_k < 1 \). This will help us compare \( a_k^2 \) to \( a_k \) for large \( k \).
3. For \( k > N \), since \( 0 < a_k < 1 \), we have \( a_k^2 < a_k \). This inequality allows us to use the Comparison Test for series convergence.
4. Apply the Comparison Test: since \( \sum a_k \) converges and \( a_k^2 < a_k \) for all \( k > N \), the tail \( \sum_{k=N+1}^\infty a_k^2 \) converges by comparison.
5. Finally, since the finite sum \( \sum_{k=1}^N a_k^2 \) is finite and the tail \( \sum_{k=N+1}^\infty a_k^2 \) converges, the entire series \( \sum a_k^2 \) converges.

Key Concepts

Convergent Series of Positive Terms

A series ∑aₖ with positive terms converges if the sequence of partial sums approaches a finite limit. Since all terms are positive, the partial sums form a non-decreasing sequence bounded above, ensuring convergence. Understanding this helps establish the behavior of the original series.

Comparison Test for Series

The comparison test states that if 0 ≤ aₖ ≤ bₖ for all k and ∑bₖ converges, then ∑aₖ also converges. This test is useful to compare the series of squares ∑aₖ² with the original series ∑aₖ or another known convergent series to prove convergence.

Behavior of Squares of Terms in a Convergent Series

If ∑aₖ converges with positive terms, then aₖ → 0 as k → ∞. For sufficiently large k, aₖ < 1, so aₖ² < aₖ. This inequality allows us to use the comparison test to show that ∑aₖ² converges by comparing it to ∑aₖ.