Exercise · ZW-ABSOLUTE-CONVERGENCE-010 · ★★★ Advanced
If \(\sum a_n\) and \(\sum b_n\) converge absolutely, what can be said about \(\sum(2a_n-3b_n)\)?
Choose one option.
Show answer
- D. It converges absolutely.
Explanation
By the triangle inequality, \(|2a_n-3b_n|\le2|a_n|+3|b_n|\), and the series on the right converges. By comparison, \(\sum(2a_n-3b_n)\) 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.
\[\sum_{n=1}^{\infty}|a_n|<\infty\]
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