Sequences and Series¶
Recall (doc/postdoc). Not a first course.
Job: completeness of \(\mathbb{R}\) via Cauchy sequences; series as limits of partial sums; power series radii.
Core¶
Sequence \(a_n\to L\): \(\forall\varepsilon\ \exists N:\ n\ge N\Rightarrow|a_n-L|<\varepsilon\). Cauchy \(\Leftrightarrow\) convergent in \(\mathbb{R}\).
Series \(\sum a_n\) converges iff \(S_N=\sum_{n=1}^N a_n\) converges.
| Test | Criterion (positive \(a_n\) unless noted) |
|---|---|
| Term test | \(a_n\not\to0\Rightarrow\) diverge |
| Comparison / limit cmp | bound by known series |
| Ratio / root | $L=\lim |
| Integral | compare to \(\int f\) |
| Alternating (Leibniz) | \(b_n\downarrow0\Rightarrow\sum(-1)^n b_n\) conv |
| Absolute | $\sum |
Power series: \(R=1/\limsup|c_n|^{1/n}\); converges for \(|x-a|<R\).
p-series: \(\sum n^{-p}\) conv iff \(p>1\). Harmonic diverges.
Used later for¶
- Real analysis: uniform convergence, Weierstrass M
- Complex analysis: holomorphic = local power series
- Fourier: series in \(L^2\)
- Functional analysis: Banach completeness
Check¶
Does \(\sum(-1)^n/\sqrt{n}\) converge absolutely? conditionally?
Not absolutely (\(p=1/2\le1\)). Conditionally yes by alternating series test.