Gregory Leibniz Calculator
Free online gregory leibniz calculator. Enter your values and get instant results with the formula, step-by-step working, a worked example and FAQs — everything runs locally in your browser on Calcton, no signup required.
中文版：格雷戈里-莱布尼茨级数在线计算器
原理：莱布尼茨 1674 年发现 π/4 = 1 − 1/3 + 1/5 − 1/7 + ⋯，即 arctan(1) 的麦克劳林展开；它是交错级数，因此部分和永远在真值两侧摆动，误差不超过下一个被舍弃项 1/(2n+3)。
步骤：① 输入求和项数 n；② 计算部分和并乘 4 得 π 估计；③ 对比误差随 n 的衰减速度。
示例：n = 100 时 4·S ≈ 3.151493401071，误差约 0.0099；n = 1000 时误差约 0.000999——项数扩大 10 倍误差只缩小 10 倍，收敛极慢。
注意事项：要得到 6 位正确数字需要约 50 万项，实际计算 π 应使用尼拉坎塔级数、马青公式或 AGM 迭代；本工具的价值在于直观展示条件收敛级数的慢收敛特性。
相关：尼拉坎塔级数同项数下误差小一个数量级；莱布尼茨判别法（207 批）解释了该级数为何保证收敛。
How to use the Gregory Leibniz Calculator
- Fill in the inputs with your own numbers — every field shows what it expects.
- The result updates instantly as you type; no "Calculate" click needed.
- Read the result card for the main figure, the step-by-step formula, and practical notes.
Frequently asked questions
Is the Gregory Leibniz Calculator free?
Yes — all math calculators on Calcton are free, with no sign-up required. Your inputs are processed in your browser and never sent to a server.
Is there a Chinese version of this tool?
Yes. Use the Chinese version link above — it is the same calculator with the interface in Simplified Chinese.
References
Formulas and benchmarks used by this calculator follow these published sources.
KevinContent LeadCalcton Editorial Team · Last updated 2026-10-03
Kevin maintains Calcton's formula verification and content: every calculator cites the standard it implements, worked examples are recomputed programmatically, and health benchmarks follow WHO and international guidelines.
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