Calculus Small Changes, Big Ideas

Section 22.1 Convergence of sequences

• A sequence of real numbers is a function \(a:\bb N\to\bb R\text{.}\) We write \(a_{n}\) instead of \(a(n)\text{.}\)

  🔗
  🔗
• A sequence is bounded if there exists \(L\) and \(U\) such that \(L\le a_{n}\le U\) for all \(n\text{.}\)

  🔗
  🔗
• A sequence converges to \(a\in\bb R\) if for all \(\eps>0\) there exists some \(N\in\bb N\) such that \(|a_{n}-a|<\eps\) for all \(n>N\text{.}\) We say \(a\) is the limit of \((a_{n})\text{,}\) and we write \(\limit{n\to\infty}a_{n}=a\text{.}\)

  🔗
  🔗
• If a sequence does not converge, it diverges.

  🔗
  🔗
• We say \(\limit{n\to\infty}a_{n}=\infty\) if for all \(M>0\) there exists some \(N\in\bb N\) such that \(a_{n}>M\) for all \(n>N\text{.}\) Similarly, \(\limit{n\to\infty}a_{n}=\infty\) if for all \(M<0\) there exists some \(N\in\bb N\) such that \(a_{n}<M\) for all \(n>N\text{.}\)

  🔗
  🔗
• A point \(x\) is a limit point of a set \(A\) if there is a sequence of points \(a_{1},a_{2},a_{3},\cdots\) from \(A\setminus\set{x}\) such that \(a_{n}\to x\text{.}\) (This seems off... not that general. Look up again.)

  🔗
  🔗
• A set is closed if and only if it contains all its limit points.

  🔗
  🔗