Calculus Small Changes, Big Ideas

Section 25.1 Convergence of series

• The infinite series \(\DS\sum_{k=1}^{\infty} a_{k}\) is defined as the limit of the sequence of partial sums (if it exists):

  \begin{equation*} \sum_{k=1}^{\infty} a_{k}=\limit{n\to\infty}\sum_{k=1}^{n} a_{k} \end{equation*}

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• Series limit laws

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• Divergence test: If \(a_{k}\not\to 0\text{,}\) then \(\DS\sum a_{k}\) diverges.

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• If a series has all nonnegative terms, then either it converges or it diverges to infinity.

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• Comparison test: Assume \(0\le a_{k}\le b_{k}\) for all \(k\text{.}\)

  • If \(\DS\sum b_{k}\) converges, then \(\DS\sum a_{k}\) converges.

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  • If \(\DS\sum a_{k}\) diverges, then \(\DS\sum b_{k}\) diverges.

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• The harmonic series diverges.

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• Integral test: If \(a_{n}=f(n)\text{,}\) then \(\DS\sum a_{n}\) converges if and only if \(\DS\int_{1}^{\infty} f(x)\dd{x}\) converges.

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• \(p\)-series test: \(\DS\sum \frac{1}{k^{p}}\) converges if and only if \(p>1\text{.}\)

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• Limit comparison test: Assume \(0\le a_{k}\le b_{k}\) for all \(k\text{.}\) If

  \begin{equation*} \limit{k\to\infty}\dfrac{a_{k}}{b_{k}} \end{equation*}

  where \(0<c<\infty\text{,}\) then \(\DS\sum a_{k}\) converges if and only if \(\DS\sum b_{k}\) converges.

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