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Convergence
convergence
Problem-solving strategies for convergence in real analysis
SKILL.md
Full skill instructions
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
- Ratio test:
-
Verify Bounds
- Use
z3_solve.py provefor inequality bounds - Check monotonicity with derivatives
- Use
-
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.
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