MATHEMATICAL ANALYSIS 02
Sequences and Convergence
A sequence converges when its terms eventually become arbitrarily close to one fixed number.
Sequences
A sequence is an ordered list \((a_n)\). For example, \(a_n=1/n\) gives \(1,1/2,1/3,\ldots\).
Convergence
\[\boxed{a_n\to L}\]
means: for every \(\varepsilon\gt0\), there is an integer \(N\) such that
\[n\ge N\quad\Longrightarrow\quad |a_n-L|\lt\varepsilon.\]The tolerance \(\varepsilon\) may be as small as we wish; sufficiently late terms must lie inside it.
Example
For \(a_n=1/n\), choose \(N\gt1/\varepsilon\). Then \(n\ge N\) gives \(1/n\lt\varepsilon\), so \(1/n\to0\).
Divergence
A sequence that does not approach a finite limit diverges. For example \((-1)^n\) oscillates between \(-1\) and \(1\).
Useful theorem
Every convergent sequence is bounded. Also, every bounded monotone sequence converges.