3.5 Sequences
Definition 3.32. A sequence of complex numbers is an ordered list of complex numbers such that corresponding to each natural number \(n\) there is an \(n^{\text {th}}\) number in the list \(a_n\). Denoted by \((a_n),\,\big \{a_n\big \}^{\infty }_{n = 1}\)
Definition 3.33. A sequence \((a_n)\) is said to have a limit \(L\), if given \(\varepsilon > 0\) there is a corresponding
\(n_0 \in \mathbb {N} \ni \forall n\,\geq n_0, \, \left |a_n - L\right | < \varepsilon \).
Definition 3.34. A sequence is said to be convergent if it has a limit and is said to be “not
convergent” if it has no limit.
A sequence is said to diverge if given \(M> 0\) there is an \(\, n_0 \in \mathbb {N} \ni \forall n\, \geq n_0 \, \left |a_n\right | > M\).
Cauchy’s Criterion
A sequence \((a_n)\) is convergent in \(\mathbb {C}\) if and only if given \(\varepsilon > 0\) there is an
\(\, n_0 \in \mathbb {N} \ni \forall n, m\geq n_0, \, \left |a_m - a_n\right | < \varepsilon \).
Bolzano - Weistrass Theorem
An infinite subset of a compact set has a limit point.
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.