Exercise · ZW-ABSOLUTE-CONVERGENCE-006 · ★★☆ Intermediate
Which argument shows that \(\sum \frac{\sin n}{n^2}\) converges absolutely?
Choose one option.
Show answer
- B. \(\left|\frac{\sin n}{n^2}\right|\le\frac1{n^2}\) and the comparison test.
Explanation
Since \(|\sin n|\le 1\), each term in absolute value is at most \(1/n^2\), whose series converges. A term tending to 0 is not enough, and \(\sin n\) is neither monotone nor regularly alternating.
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?
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- 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-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?