What if multiple tests are inconclusive?

Try a bounding comparison or assess monotonicity and bounds, then apply another test.

How do I compute the sum after proving convergence?

Use the symbolic sum command to evaluate the series once convergence is established.

Which tools are recommended for limits and sums?

Use Sympy_Limit for limits, Sympy_Sum for sums, and Z3_Prove for bounding inequalities.

convergence

npx machina-cli add skill parcadei/Continuous-Claude-v3/convergence --openclaw

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SKILL.md

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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.

Source

git clone https://github.com/parcadei/Continuous-Claude-v3/blob/main/.claude/skills/math/real-analysis/convergence/SKILL.md

View on GitHub

Overview

This skill guides you through identifying sequence or series type, selecting convergence tests, and bounding terms. It also covers computing sums when a series converges using symbolic tools.

How This Skill Works

Classify the sequence or series as geometric, p series, or alternating. Then apply ratio, root, or comparison tests and verify bounds with solver tools. If convergent, compute the sum using symbolic summation.

When to Use It

Determining convergence of a geometric series with |r|<1
Testing a p-series for convergence when p>1
Assessing convergence of an alternating series with decreasing terms to zero
Bounding a series with inequalities and using Z3_Prove to verify limits
Computing the sum of a convergent series with symbolic summation

Quick Start

Step 1: Identify the sequence or series type (geometric, p-series, alternating)
Step 2: Apply a convergence test (ratio, root, or comparison) and verify bounds with tools as needed
Step 3: If convergent, compute the sum with Sympy_Sum or prove bounds with Z3_Prove

Best Practices

Identify the sequence or series type up front
Choose the most appropriate convergence test for the form
Check monotonicity and bounds, using derivatives when helpful
Use symbolic tools to verify limits and compute sums when convergent
Cross-check results with alternative tests or bounding comparisons

Example Use Cases

Geometric series with |r|<1 and its sum
p-series with p>1 and its convergence
Alternating series with decreasing terms to zero
Convergence proofs by comparison to bounding series
Exact sums obtained via symbolic computation

Frequently Asked Questions

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