Sequences and Series
Sequences
A sequence \({a_n}\) converges to L if \(\lim_{n\to\infty}a_n=L\).
Series
\(\sum_{n=1}^\infty a_n\) converges if partial sums converge. Geometric series: \(\sum ar^n=\frac{a}{1-r}\) for |r|<1.
Convergence Tests
| Property | Statement |
|---|---|
| Ratio | \(L=\lim|a_{{n+1}}/a_n|\): converges if L<1 |
| Integral | Compare to \(\int f\) |
| Comparison | Compare to known series |
Examples
Example 1. Does \(\sum 1/n^2\) converge?
Solution. Yes, p-series with p=2>1.
Deep Dive: Sequences And Series
This section builds durable understanding of sequences and series in calculus through definition-first reasoning, theorem mapping, and error-checking workflows.
Use a two-pass method: first derive the structure symbolically, then validate with a concrete numerical or geometric test case.
Visual Intuition
Convert algebra into a diagram, graph, or dependency map before solving. Visual-first analysis reduces sign errors and makes assumptions explicit.
Practice Set
Practice A. Re-derive one key formula on this page from first principles and annotate each transformation.
Target. Your final line should include assumptions, derivation path, and a quick verification.
Practice B. Build an application scenario using sequences and series and solve it with both symbolic and numeric methods.
Target. Compare outputs and explain any approximation gap.
References & Editorial Notes
- Stewart, Calculus.
- Strang, Introduction to Linear Algebra.
- Apostol, Mathematical Analysis.
Editorial update: Reviewed on 2026-04-14 for notation consistency, conceptual clarity, and exercise quality.