convergence

# 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.