Calculus Small Changes, Big Ideas

Section 25.2 Absolute and conditional convergence

• Alternating series test: If the alternating series

  \begin{equation*} \sum_{k=1}^{\infty} (-1)^{k-1}b_{k}=b_{1}-b_{2}+b_{3}-b_{4}+\cdots\quad(b_{k}>0) \end{equation*}

  satisfies the conditions \(b_{k+1}\le b_{k}\) and \(b_{n}\to 0\text{,}\) then the series converges.

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• \(\displaystyle |R_{n}|=|s-s_{n}|\le b_{n+1}\)

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• A series \(\DS\sum a_{k}\) is called absolutely convergent if the series \(\DS\sum |a_{k}|\) converges. (Absolute convergence implies convergence.)

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• A series \(\DS\sum a_{k}\) is called conditionally convergent if \(\DS\sum a_{k}\) converges but \(\DS\sum |a_{k}|\) diverges.

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• Riemann rearrangement theorem: If \(\DS\sum a_{k}\) is a conditionally convergent series and \(r\) is any real number (or \(\infty\) or \(-\infty\)), then there is a rearrangement of \(\DS\sum a_{k}\) that has a sum equal to \(r\text{.}\)

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• Ratio test: Let \(L=\limit{k\to\infty}\abs{\dfrac{a_{k+1}}{a_{k}}}\text{.}\)

  • If \(L<1\text{,}\) then \(\DS\sum a_{k}\) converges absolutely.

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  • If \(L>1\text{,}\) then \(\DS\sum a_{k}\) diverges.

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  • If \(L=1\text{,}\) the test is inconclusive.

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• Root test: Let \(L=\limit{k\to\infty}\sqrt[k]{\abs{a_k}}\text{.}\)

  • If \(L<1\text{,}\) then \(\DS\sum a_{k}\) converges absolutely.

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  • If \(L>1\text{,}\) then \(\DS\sum a_{k}\) diverges.

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  • If \(L=1\text{,}\) the test is inconclusive.

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