| 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
-
Identify Sequence/Series Type - Geometric series: |r| < 1 converges - p-series: p > 1 converges - Alternating series: check decreasing + limit 0
-
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 -
Verify Bounds - Use
z3_solve.py provefor inequality bounds - Check monotonicity with derivatives -
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.