Numerical Sequences and Series¶
Convergent Sequences¶
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Definition (Convergent Sequence): A sequence \(\{p_n\}\) in a metric space \(X\) is convergent if \(\exists p \in X\) such that \(\forall \varepsilon > 0, \exists N\) with \(n \ge N \implies d(p_n,p) < \varepsilon\). We write \(p_n \to p\) or \(\lim_{n\to\infty} p_n = p\).
Tip
With every step you take, you get closer to your destination. After a certain number of steps, you are guaranteed to be within \(\varepsilon\) of the destination. The destination is within reach.
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Definition (Range of a Sequence): The set of all points \(p_n\) is the range of \(\{p_n\}\).
Sequences in general metric space¶
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Theorem: Let \(\{p_n\}\) be a sequence in a metric space \(X\).
\(\{p_n\} \to p \in X\) iff every neighbourhood of \(p\) contains \(p_n\) for all but finitely many \(n\).
\((\{p_n\} \to p)\wedge(\{p_n\} \to p') \implies p=p'\).
If \(\{p_n\}\) converges, then \(\{p_n\}\) is bounded.
If \(E \subseteq X\) and \(p\) is a limit point of \(E\), then there exists a sequence \(\{p_n\}\) in \(E\) such that \(\{p_n\} \to p\).
Sequences in \(\mathbb{C}\)¶
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Theorem: Let \(\{s_n\}\) and \(\{t_n\}\) be complex sequences such that \(\{s_n\}\to s\) and \(\{t_n\}\to t\). Then
\(\{s_n+t_n\} \to s+t\).
\(\{s_n\cdot t_n\} \to s\cdot t\).
\(\{c\cdot s_n\} \to c\cdot s\).
\(\{1/s_n\} \to 1/s\) where \(s_n\neq 0\) and \(s\neq 0\).
Sequences in \(\mathbb{R}^k\)¶
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Theorem: Let \(x_n\in \mathbb{R}^k\) and \(x_n=(\alpha_{1,n},\cdots,\alpha_{k,n})\) and \(x=(\alpha_1,\cdots,\alpha_k)\). Then \(\{x_n\}\to x\) iff \(\alpha_{j,n}\to \alpha_j\) for \(1\le j\le k\).
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Theorem: Suppose \(\{x_n\}\) and \(\{y_n\}\) are sequences in \(\mathbb{R}^k\) and \(\{\beta_n\}\) is a sequence of real numbers such that \(\{x_n\}\to x\), \(\{y_n\}\to y\), \(\{\beta_n\}\to \beta\). Then
\(\{x_n+y_n\}\to x+y\).
\(\{x_n\cdot y_n\}\to x\cdot y\).
\(\{\beta_n\cdot x_n\}\to \beta\cdot x\).
Subsequences¶
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Definition (Subsequence): Given a sequence \(\{p_n\}\), consider a sequence of positive integers \(\{n_k\}\) with \(n_1<n_2<\cdots\). Then \(\{p_{n_k}\}\) is a subsequence of \(\{p_n\}\).
Tip
Imagine someone tracing your steps but choosing to hop and skip a few footprints as they wish.
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Definition (Subsequential limit): If \(\{p_{n_k}\}\to p\), then \(p\) is a subsequential limit of \(\{p_n\}\).
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Theorem: Every sequence in a compact metric space \(K\) has a subsequence which converges to some point in \(K\).
Theorem: Every bounded sequence in \(\mathbb{R}^k\) has a subsequence which converges to some point in \(\mathbb{R}^k\).
Theorem: The subsequential limits of a sequence \(\{p_n\}\) in a metric space \(X\) form a closed subset of \(X\).
Cauchy Sequences¶
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Definition (Cauchy Sequence): A sequence \(\{p_n\}\) in a metric space \(X\) is Cauchy if \(\forall \varepsilon>0,\ \exists N\) such that \(\forall n,m\ge N,\ d(p_n,p_m)<\varepsilon\).
Tip
With every step you take, your step-size gets smaller. After enough steps, any two future steps are within \(\varepsilon\) of each other. The final destination may still be out of reach.
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Definition (Diameter): For \(E\subset X,\ E\neq \varnothing\), \(\mathrm{diam}\,E := \sup\{d(p,q)\mid p,q\in E\}\).
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Theorem: Let \(E^{\circ}\) be the closure of a set \(E\subset X\), where \(X\) is a metric space. Then \(\mathrm{diam}\,E^{\circ}=\mathrm{diam}\,E\).
Theorem: Let \(K_n\) be a sequence of compact sets in \(X\) such that \(K_n \supset K_{n+1}\).
\[\mathrm{diam}\,K_n \to 0 \ \Longrightarrow\ \bigcap_{n=1}^{\infty} K_n\ \text{is singleton.}\]Theorem: In any metric space \(X\), \(\{p_n\}\) convergent \(\Longrightarrow\) \(\{p_n\}\) is Cauchy.
Theorem: Every Cauchy sequence in a compact metric space \(K\) converges to some point in \(K\).
Theorem: Every Cauchy sequence in \(\mathbb{R}^k\) converges to some point in \(\mathbb{R}^k\).
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Definition (Complete Metric Space): A metric space where every Cauchy sequence converges to some point is complete.
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Definition (Monotonic): A sequence \(\{s_n\}\) of real numbers is
monotonically increasing if \(s_n \le s_{n+1}\),
monotonically decreasing if \(s_n \ge s_{n+1}\).
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Theorem: Suppose \(\{s_n\}\) is monotonic. \(\{s_n\}\) converges \(\Longleftrightarrow\) \(\{s_n\}\) is bounded.
Upper and Lower Limits¶
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Definition (\(s_n\to +\infty\)): Let \(\{s_n\}\) be a sequence of real numbers, such that \(\forall M\in\mathbb{R},\exists N\in \mathbb{Z}^{+}\) with \(n\ge N \implies s_n\ge M\). Write \(s_n\to +\infty\).
Definition (\(s_n\to -\infty\)): Let \(\{s_n\}\) be a sequence of real numbers, such that \(\forall M\in\mathbb{R},\exists N\in \mathbb{Z}^{+}\) with \(n\ge N \implies s_n\le M\). Write \(s_n\to -\infty\).
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Definition (Upper and Lower Limits): Let \(\{s_n\}\) be a sequence of real numbers. Let \(E\) be the set of numbers \(x\in \mathbb{R}_{\{-\infty,+\infty\}}\) such that \(s_{n_k}\to x\) for some subsequence \(\{s_{n_k}\}\). This set \(E\) contains all subsequential limits and possibly \(-\infty\) and \(+\infty\). Define \(s^{*}=\sup E\) and \(s_{*}=\inf E\) as the upper-limit and lower-limit of \(\{s_n\}\):
\[\limsup_{n\to\infty} s_n = s^{*}\]\[\liminf_{n\to\infty} s_n = s_{*}.\]
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Theorem: For a sequence of real numbers \(\{s_n\}\):
\(s^{*}, s_{*}\in E\).
If \(x>s^{*}\), \(\exists N\in \mathbb{Z}^{+}\) such that \(n\ge N \implies s_n < x\).
If \(x<s_{*}\), \(\exists N\in \mathbb{Z}^{+}\) such that \(n\ge N \implies s_n > x\).
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Theorem: If \(s_n \le t_n\) for \(n\ge N\) (some fixed \(N\)), then
\(\displaystyle \liminf_{n\to\infty} s_n \le \liminf_{n\to\infty} t_n\),
\(\displaystyle \limsup_{n\to\infty} s_n \le \limsup_{n\to\infty} t_n\).
Some Special Sequences¶
The following limits are commonly used:
\(\displaystyle \lim_{n\to\infty}\frac{1}{n^{p}}=0\), where \(p>0\).
\(\displaystyle \lim_{n\to\infty} n^{1/p}=1\), where \(p>0\).
\(\displaystyle \lim_{n\to\infty} n^{1/n}=1\).
\(\displaystyle \lim_{n\to\infty} (n!)^{1/n}=+\infty\).
\(\displaystyle \lim_{n\to\infty} a^{n}=0\), where \(|a|<1\).
\(\displaystyle \lim_{n\to\infty} a^{1/n}=1\).
\(\displaystyle \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^{n}=e\).
\(\displaystyle \lim_{n\to\infty} \left(1+\frac{a}{n}\right)^{bn}=e^{ab}\).
\(\displaystyle \lim_{n\to\infty} \arctan(n)=\pi/2\).
\(\displaystyle \lim_{n\to\infty} \frac{n^{\alpha}}{a^{n}}=0\), where \(a>1\) and \(\alpha\in\mathbb{R}\).
\(\displaystyle \lim_{n\to\infty} \frac{P(n)}{e^{n}}=0\), where \(P(n)\) is a polynomial.
\(\displaystyle \lim_{n\to\infty} \frac{\log(n)}{n^{a}}=0\), where \(a>0\).
\(\displaystyle \lim_{n\to\infty} \frac{a^{n}}{n!}=0\).
Series¶
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Series of Nonnegative Terms¶
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The Number \(e\)¶
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The Root and Ratio Tests¶
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Power Series¶
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Summation By Parts¶
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Absolute Convergence¶
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Addition and Multiplication of Series¶
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Rearrangements¶
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