Exercise · ZW-ABSOLUTE-CONVERGENCE-009 · ★★★ Advanced
If \(\sum a_n\) converges absolutely, what happens to \(\sum a_n^2\)?
Choose one option.
Show answer
- A. It converges, because \(a_n^2\le|a_n|\) from some index on.
Explanation
Since \(a_n\to0\), there is \(N\) with \(|a_n|\le1\) for \(n\ge N\), so \(a_n^2\le|a_n|\). Compare with \(\sum|a_n|\). The sum of squares is not the square of the sum.
Review the concept: Absolute convergence
A series \(\sum a_n\) of real numbers converges absolutely when the series formed by the absolute values of its terms converges.
\[\sum_{n=1}^{\infty}|a_n|<\infty\]
More practice problems: absolute convergence
- ZW-ABSOLUTE-CONVERGENCE-001 Which of these series converges absolutely?
- ZW-ABSOLUTE-CONVERGENCE-002 Compute ∑_n=0^∞|(-tfrac12)^n|.
- ZW-ABSOLUTE-CONVERGENCE-003 If ∑ |a_n| converges, then ∑ a_n…
- ZW-ABSOLUTE-CONVERGENCE-004 The alternating harmonic series ∑_n=1^∞((-1)^n+1)/(n) converges to ln 2. How does it converge?
- ZW-ABSOLUTE-CONVERGENCE-005 For which values of p does ∑ ((-1)^n)/(n^p) converge absolutely?
- ZW-ABSOLUTE-CONVERGENCE-006 Which argument shows that ∑ (sin n)/(n^2) converges absolutely?
- ZW-ABSOLUTE-CONVERGENCE-007 Compute ∑_n=1^∞|((-1)^n)/(3^n)|.
- ZW-ABSOLUTE-CONVERGENCE-008 Which of the following statements is true?