Exercise · ZW-ABSOLUTE-CONVERGENCE-003 · ★☆☆ Basic
If \(\sum |a_n|\) converges, then \(\sum a_n\)…
Choose one option.
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- B. converges.
Explanation
Every absolutely convergent series converges: this follows from the Cauchy criterion and the inequality \(|a_{m+1}+\dots+a_n|\le |a_{m+1}|+\dots+|a_n|\).
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-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?
- ZW-ABSOLUTE-CONVERGENCE-009 If ∑ a_n converges absolutely, what happens to ∑ a_n^2?