Sequences
These object are needed to define limits later on.
Discover Bridges
Study Sequences
Definition 1. Let \(M\) be a set. A sequence in \(M\) is a map \(a:\mathbb{N}\to M\) or \(a:\mathbb{N}_0 \to M\).
We use the following symbols for sequences: \[(a_n)_{n\in\mathbb{N}},\qquad (a_n),\qquad (a_n)_{n=1}^\infty,\quad (a_1,a_2,a_3,\ldots).\]
Remark 2. \(M\) is usually a real subset (\(M\subset\mathbb{R}\)), but \(M\) can also be a complex subset (\(M\subset\mathbb{C}\)) or a subset of some normed space (or \(M=\mathbb{R}\), \(M=\mathbb{C}\), \(M\) a normed space itself).
Example 3.
\(a_n=(-1)^n\), then \((a_n)_{n\in\mathbb{N}}=((-1)^n)_{n\in\mathbb{N}}=(-1,1,-1,1,-1,1,\ldots)\);
\(a_n=\frac1n\), then \((a_n)_{n\in\mathbb{N}}=(\frac1n)_{n\in\mathbb{N}}=(1,\frac12,\frac13,\frac14,\frac15,\frac16,\ldots)\);
\(a_n=\mathrm{i}^n\) (\(\mathrm{i}\) is the imaginary unit), then \((a_n)_{n\in\mathbb{N}}=(\mathrm{i}^n)_{n\in\mathbb{N}}=(\mathrm{i},-1,-\mathrm{i},1,\mathrm{i},-1,\ldots)\);
\(a_n=\frac1{2^n}\), then \((a_n)_{n\in\mathbb{N}}=(\frac1{2^n})_{n\in\mathbb{N}}=(1,\frac12,\frac14,\frac18,\frac1{16},\frac1{32},\ldots)\);
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