Exercise · ZW-ABSOLUTE-CONVERGENCE-004 · ★★☆ Intermediate
The alternating harmonic series \(\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\) converges to \(\ln 2\). How does it converge?
Choose one option.
Show answer
- C. It converges, but not absolutely.
Explanation
It converges by Leibniz's test, but the series of absolute values is the harmonic series, which diverges. This is conditional convergence.
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-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?