Series in a Normed Space

Section 8.3 Series in a Normed Space

We haven’t talked at all about series so far. That’s because a series is nothing more than a particular kind of sequence.

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Definition 8.3.1.

Given a sequence \(v_n \in V\) in a normed space, the series is the sequence

\begin{equation*} S_N=\displaystyle\sum_{n=1}^N v_n\ \ . \end{equation*}

We say that a series converges to \(S\in V\) if \(S_N\to S\text{,}\) and we notate this notion by writing

\begin{equation*} S=\displaystyle\sum_{n=1}^\infty v_n\ \ \ . \end{equation*}

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Checkpoint 8.3.2.

Definition 8.3.1 makes clear that every series is a sequence. But it’s also true that every sequence is a series. Prove this by, for a given sequence \((x_n)_{n\in\mathbb{N}}\text{,}\) finding a sequence \((v_n)_{n\in\mathbb{N}}\) with \(\displaystyle x_N=\sum_{n=1}^N v_n\text{.}\)

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This proposition records some nice facts about series, each of which is just a basic arithmetic fact and a fact about limits of sequences.

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Proposition 8.3.3.

If \(\displaystyle A=\sum_{k=0}^\infty a_k\) and \(\lambda\in \mathbb{R}\text{,}\) then \(\displaystyle A=\sum_{k=0}^\infty \lambda a_k\) converges and we have
\begin{equation*} \displaystyle\sum_{k=0}^\infty \lambda a_k=\lambda A \end{equation*}

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If \(\displaystyle A=\sum_{k=0}^\infty a_k\) and \(\displaystyle B=\sum_{k=0}^\infty b_k\text{,}\) then \(\displaystyle \sum_{k=0}^\infty(a_k+b_k)\) converges, and we have
\begin{equation*} \displaystyle \sum_{k=0}^\infty(a_k+b_k)=A+B \end{equation*}

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If \(\displaystyle A=\sum_{k=0}^\infty a_k\) and \(\displaystyle B=\sum_{k=0}^\infty b_k\text{,}\) and \(\forall k, a_k\leq b_k\text{,}\) then \(A\leq B\text{.}\)

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Proposition 8.3.4. Cauchy criterion for series.

Let \(V\) be a Cauchy-complete normed space. \(\displaystyle \sum_{k=0}^\infty a_n\) converges if and only if for any \(\epsilon\gt 0\text{,}\) there is \(N\in\mathbb{N}\) so that for any \(m,n\gt N\text{,}\) \(\displaystyle \left\lVert \sum_{k=n}^m a_k\right\rVert\lt \epsilon\text{.}\)

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Corollary 8.3.5. Two Officers and a Drunk for Series.

Consider three series of real numbers \(\displaystyle\sum_{k=0}^\infty a_k,\sum_{k=0}^\infty b_k, \sum_{k=0}^\infty c_k\text{,}\) with the property that \(\forall k, a_k\leq b_k\leq c_k\text{.}\) If \(\displaystyle\sum_{k=0}^\infty a_k\) and \(\displaystyle\sum_{k=0}^\infty c_k\) converge, then so does \(\displaystyle\sum_{k=0}^\infty b_k\text{.}\)

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Checkpoint 8.3.6.

There’s an important difference between Corollary 8.3.5 and its sequential counterpart Proposition 2.2.7, which makes the series formulation seem rather more miraculous. Identify this difference.

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Corollary 8.3.7.

If \(\displaystyle \sum_{k=0}^\infty \lVert v_k\rVert\) converges, then so does \(\displaystyle \sum_{k=0}^\infty v_k\text{;}\) moreover, we have

\begin{equation*} \displaystyle\left\lVert \sum_{k=0}^\infty v_k\right\rVert \leq \sum_{k=0}^\infty \lVert v_k\rVert\ \ \ . \end{equation*}

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Observe that the two series in Corollary 8.3.7 occur in different spaces: one of them is a series of real numbers; the other is a series in a normed space.

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Theorem 8.3.8. Root Test.

Define \(L=\limsup \sqrt[k]{\lVert a_k\rVert}\text{.}\)

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If \(L\lt 1\text{,}\) then \(\displaystyle \sum_{k=0}^\infty \lVert a_k\rVert\) converges (hence, \(\displaystyle \sum_{k=0}^\infty a_k\) converges.)

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If \(L\gt 1\text{,}\) then \(\displaystyle \sum_{k=0}^\infty \lVert a_k\rVert\) diverges.

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Proof.

Notice that the statements here are really statements about series of nonnegative real numbers. So we’ll assume from here on out that each \(a_k\in[0,\infty)\text{.}\)

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First, let’s handle the case \(L\lt 1\text{.}\) The hypothesis implies that there is \(K\in\mathbb{N}\) so that \(k\gt K\) guarantees \(a_k\leq \left(\frac{L+1}{2}\right)^k\text{.}\) Now the series

\begin{equation*} \displaystyle \sum_{k=0}^\infty r^k \end{equation*}

converges for any \(r\in [0,1)\text{.}\) So Two Officers and a Drunk, applied with \(0\leq a_k\leq \left(\frac{L+1}{2}\right)^k\) gives the result.

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On the other hand, if \(L\gt 1\text{,}\) the hypothesis implies that there are infinitely many \(n\in \mathbb{N}\) with \(a_n\geq \left(\frac{L+1}{2}\right)^n\text{.}\) These form a subsequence \(a_{n_k}\) with \(a_{n_k}\to \infty\text{,}\) which by Proposition 8.3.4 means \(\displaystyle\sum_{k=0}^\infty a_k\) cannot converge.

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Here’s how I view the theory of numerical series: the only series whose convergence is really well understood is the geometric series \(\displaystyle\sum_{k=0}^\infty r^k\text{,}\) and all other series we compare to a geometric series using the Root Test. This is, of course, extremely reductive -- but for our purposes, it works well enough.

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