Exercise · ZW-ABSOLUTE-CONVERGENCE-008 · ★★☆ Intermediate
Which of the following statements is true?
Choose one option.
Show answer
- C. If \(\sum a_n\) converges absolutely, then \(a_n\to 0\).
Explanation
If \(\sum|a_n|\) converges, its terms tend to 0, hence \(a_n\to0\). The other three fail: the alternating harmonic series refutes “if it converges, it converges absolutely” and “if \(\sum|a_n|\) diverges, then \(\sum a_n\) diverges”, and \(a_n=1/n\) refutes “if \(a_n\to 0\), it converges absolutely”.
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-009 If ∑ a_n converges absolutely, what happens to ∑ a_n^2?