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

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.