Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
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Section 3.1 Infinite Series Basics
“The whole is greater than the sum of its parts.”
―Aristotle
Definition 3.1 .
A series is a sequence \((\sum_{k=1}^na_k)_{n=1}^{\infty}\) for some sequence \((a_k)_{k=1}^{\infty}\subseteq \mathbb{C}\text{.}\) We'll typically write the series as \(\sum_{k=1}^{\infty}a_k\text{.}\) We say \(\sum_{k=1}^{\infty}a_k\) converges/diverges when \((\sum_{k=1}^na_k)_{n=1}^{\infty}\) converges/diverges. When it converges we write \(\sum_{k=1}^{\infty}a_k=\lim_{n\rightarrow\infty}\sum_{k=1}^na_k\in\mathbb{C}\text{.}\)
Theorem 3.2 . Cauchy's Criterion.
A series \(\sum_{k=1}^{\infty}a_k\) converges iff for all \(\varepsilon>0\,\exists N\geq 1\, \forall n\geq m\geq N\) we have \(|\sum_{k=m}^n a_k|<\varepsilon\)
Proof. Cauchy's condition (RHS of iff) is equivalent to saying \((\sum_{k=1}^n a_k)_{n=1}^{\infty}\) is Cauchy.
Corollary 3.3 .
If \(\sum_{n=1}^{\infty} a_n\) converges, then \(\lim_{n\rightarrow\infty} a_n=0\text{.}\) (If limit doesn't go to zero, the sum diverges.)
