Exercise · ZW-ABSOLUTE-CONVERGENCE-005 · ★★☆ Intermediate
For which values of \(p\) does \(\sum \frac{(-1)^n}{n^p}\) converge absolutely?
Choose one option.
Show answer
- D. \(p>1\)
Explanation
The series of absolute values is \(\sum 1/n^p\), a \(p\)-series that converges exactly when \(p>1\). For \(0<p\le 1\) the convergence is only conditional.
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-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?