Vibecosystem convergence

Problem-solving strategies for convergence in real analysis

install

source · Clone the upstream repo

git clone https://github.com/vibeeval/vibecosystem

Claude Code · Install into ~/.claude/skills/

T=$(mktemp -d) && git clone --depth=1 https://github.com/vibeeval/vibecosystem "$T" && mkdir -p ~/.claude/skills && cp -r "$T/skills/math/real-analysis/convergence" ~/.claude/skills/vibeeval-vibecosystem-convergence && rm -rf "$T"

manifest: skills/math/real-analysis/convergence/SKILL.md

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 prove
```
  for 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.