AgentSkillsCN

convergence

实分析收敛问题的解题策略

SKILL.md
--- frontmatter
name: convergence
description: "Problem-solving strategies for convergence in real analysis"
allowed-tools: [Bash, Read]

Convergence

When to Use

Use this skill when working on convergence problems in real analysis.

Decision Tree

  1. •

    Identify Sequence/Series Type

    • •Geometric series: |r| < 1 converges
    • •p-series: p > 1 converges
    • •Alternating series: check decreasing + limit 0
  2. •

    Apply Convergence Tests

    • •Ratio test: sympy_compute.py limit "a_{n+1}/a_n"
    • •Root test: sympy_compute.py limit "a_n^(1/n)"
    • •Comparison test: find bounding series
  3. •

    Verify Bounds

    • •Use z3_solve.py prove for inequality bounds
    • •Check monotonicity with derivatives
  4. •

    Compute Sum (if convergent)

    • •sympy_compute.py sum "a_n" --var n --from 0 --to oo

Tool Commands

Sympy_Limit

bash
uv run python -m runtime.harness scripts/sympy_compute.py limit "a_n" --var n --at oo

Sympy_Sum

bash
uv run python -m runtime.harness scripts/sympy_compute.py sum "1/n**2" --var n --from 1 --to oo

Z3_Prove

bash
uv run python -m runtime.harness scripts/z3_solve.py prove "series_bounded"

Cognitive Tools Reference

See .claude/skills/math-mode/SKILL.md for full tool documentation.