/convergence

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

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

Sympy_Sum

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

Z3_Prove

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.