Project 14: The Sequences and Series Observatory

Build a sequence engine that computes terms, partial sums, and convergence diagnostics for arithmetic, geometric, and recursive patterns.

Quick Reference

Attribute Value
Difficulty Intermediate (Level 2)
Time Estimate 8-16 hours
Main Programming Language Python
Alternative Programming Languages Julia, R, JavaScript
Key Topics Sequences, recurrence, sigma notation, convergence
Input Mode CLI parameters and recurrence strings
Output Mode Tabular diagnostics + trend plots

1) Learning Objectives

  1. Generate arithmetic/geometric sequences from parameterized definitions.
  2. Implement recursive sequence updates and track state evolution.
  3. Compute partial sums and compare with closed-form expectations.
  4. Classify convergent vs divergent behavior with explicit criteria.
  5. Communicate limitations of finite-step convergence detection.

2) All Theory Needed (Per-Concept Breakdown)

Concept A: Term Rules vs Process Rules

A sequence can be defined explicitly (a_n) or recursively (a_{n+1} from a_n). Explicit formulas are direct access; recursion models process evolution. Both are valuable and often interchangeable for analysis.

Concept B: Partial Sums and Series

Series analysis adds accumulation behavior on top of individual terms. The distinction between term behavior and sum behavior is essential: terms can approach zero while series still diverges.

Concept C: Convergence Diagnostics

A numerical project cannot prove convergence for all n, but it can provide evidence using trend windows, tolerance thresholds, and comparison against known theoretical criteria.

3) Project Specification

3.1 What You Will Build

A CLI tool supporting:

  1. Arithmetic and geometric sequence generation.
  2. Recursive rule evaluation on fixed horizon n.
  3. Partial-sum computation and convergence status reporting.
  4. Plot exports for terms and sums.

3.2 Functional Requirements

  1. Accept sequence type and parameters.
  2. Produce first n terms and n partial sums.
  3. Report closed-form value when available.
  4. Estimate limit/fixed-point when appropriate.
  5. Save machine-readable CSV and visual plots.

3.3 Non-Functional Requirements

  • Reproducible ordering and numeric precision.
  • Clear warnings when convergence is inconclusive.
  • Stable behavior for moderately large n values.

3.4 Real World Outcome

$ python seq_observatory.py --type geometric --a1 100 --r 0.8 --n 20
[result] a20 = 1.4412
[result] S20 = 494.2354
[classification] term_convergent=true, series_convergent=true
[output] saved chart: outputs/geometric_decay_n20.png

$ python seq_observatory.py --type recursive --rule "a[n+1]=0.5*a[n]+3" --a1 2 --n 25
[estimate] fixed_point = 6.0000
[diagnostic] distance_to_fixed_point decreasing=true

4) Solution Architecture

4.1 High-Level Design

Input Parser -> Sequence Generator -> Summation Engine -> Convergence Analyzer -> Reporter

4.2 Key Components

Component Responsibility
Generator Produce terms by explicit or recursive rule
Summation Engine Compute cumulative sums with stable accumulation
Analyzer Compute trend metrics and classify behavior
Reporter Export table, plot, and interpretation summary

5) Implementation Guide

Phase 1: Deterministic Core

  • Implement arithmetic/geometric explicit generators.
  • Add partial-sum table output.
  • Add baseline tests with hand-computable examples.

Phase 2: Recursive Engine

  • Parse/encode recurrence templates.
  • Add fixed-point estimation diagnostics.
  • Compare recurrence behavior under parameter variation.

Phase 3: Convergence Reporting

  • Add convergence criteria panel (term trend, ratio trend).
  • Mark outcomes as convergent/divergent/inconclusive.
  • Generate one summary artifact per run.

6) Validation Checklist

  • Explicit and recursive sequences match expected first terms.
  • Partial sums align with manual calculations on sample cases.
  • Known convergent/divergent examples are classified correctly.
  • Reports include both numeric and visual diagnostics.

7) Extension Ideas

  1. Add telescoping and harmonic-series examples.
  2. Add symbolic sigma-expansion explanations.
  3. Add Monte Carlo perturbation of recurrence parameters.

8) Books and References

  • Concrete Mathematics - recurrences and sums.
  • Fluent Python - iteration and generator patterns.
  • High-school precalculus sequence/series chapters.