Essential Concepts
- For an alternating series [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}[/latex], if [latex]{b}_{k+1}\le {b}_{k}[/latex] for all [latex]k[/latex] and [latex]{b}_{k}\to 0[/latex] as [latex]k\to \infty [/latex], the alternating series converges.
- If [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] converges, then [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges.
Key Equations
- Alternating series
[latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}={b}_{1}-{b}_{2}+{b}_{3}-{b}_{4}+\cdots \text{or}[/latex]
[latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}=\text{-}{b}_{1}+{b}_{2}-{b}_{3}+{b}_{4}-\cdots [/latex]
Glossary
- absolute convergence
- if the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] converges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge absolutely
- alternating series
- a series of the form [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}[/latex] or [latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}[/latex], where [latex]{b}_{n}\ge 0[/latex], is called an alternating series
- alternating series test
- for an alternating series of either form, if [latex]{b}_{n+1}\le {b}_{n}[/latex] for all integers [latex]n\ge 1[/latex] and [latex]{b}_{n}\to 0[/latex], then an alternating series converges
- conditional convergence
- if the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] converges, but the series [latex]\displaystyle\sum _{n=1}^{\infty }|{a}_{n}|[/latex] diverges, the series [latex]\displaystyle\sum _{n=1}^{\infty }{a}_{n}[/latex] is said to converge conditionally
