Calculus Small Changes, Big Ideas

Section 25.5 Convergence of power series

• For a power series \(\DS\sum_{k=0}^{\infty} c_{k}(x-a)^{k}\text{,}\) there are only three possibilities:

  • The series converges only when \(x=a\text{.}\)

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  • The series converges for all \(x\text{.}\)

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  • There is a positive number \(R\) called the radius of convergence such that the series converges if \(|x-a|<R\) and diverges if \(|x-a|>R\text{.}\)

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    In this case, the interval of convergence can be open, closed, or half-open. The power series converges absolutely, since we use the Ratio Test to find it.

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• A series of functions converges pointwise/uniformly if its sequence of partial sums converges pointwise/uniformly.

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• Cauchy criterion for series of functions: Let \(f_{k}:A\to\bb R\text{.}\) The series \(\DS\sum f_{k}\) converges uniformly on \(A\) if and only if for every \(\eps>0\) there exists some \(N\in\bb N\) such that

  \begin{equation*} \abs{\sum_{k=m}^n f_k(x)}<\eps \end{equation*}

  for all \(n\ge m\ge N\) and all \(x\in A\text{.}\)

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• Weierstrass \(M\)-test: Let \(f_{k}:A\to\bb R\) and suppose that, for each \(k\in\bb N\text{,}\) there exists \(M_{k}\in\bb R\) such that \(|f_{k}(x)|\le M_{k}\) for all \(x\in A\text{.}\) If \(\DS\sum M_{k}\) converges, then \(\DS\sum f_{k}\) converges uniformly on \(A\text{.}\)

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• Power series converge uniformly on closed intervals.

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• If a power series converges absolutely at \(x=c\) it converges uniformly on the closed interval \([-|c|,|c|]\text{.}\)

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• Power series can be differentiated and integrated term-by-term.

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• Let \(E_{n}(x)=f(x)-T_{x=c}^{n}(f)\) be the error term of a finite Taylor approximation.

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• Assume \(f\) is smooth in an interval \(I\) and \(c\in I\text{.}\) Then for \(x\in I\text{,}\)

  \begin{equation*} f(x)=\sum_{k=0}^{\infty}\frac{f\psup k(c)}{k!}(x-c)^{k} \end{equation*}

  if and only if \(E_{n}(x)\to 0\) pointwise.

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• A Fourier series converges uniformly if any of these hold:

  • \(f\) satisfies a Hölder condition such as Lipschitz continuity

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  • \(f\) is continuous with bounded variation

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  • \(f\) is continuous and its Fourier coefficients are absolutely summable

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