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[提问] 七根火柴求面积

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发表于 2020-4-16 19:57:41 | 显示全部楼层
本帖最后由 王守恩 于 2020-4-16 20:08 编辑
lsr314 发表于 2020-4-10 11:36
如果火柴的长度是1,那么n根火柴围成的面积大约是$s=1/8cot(pi/(2n))$.

还能化简吗?

\(\D\sum_{k=1}^n\frac{\cos\big(\frac{(n-k+1)\pi}{2n+1}\big)\sin^3\big(\frac{(n+k)\pi}{2n+1}\big)}{2\cos\big(\frac{(k-1)\pi}{2n+1}\big)\cos\big(\frac{k*\pi}{2n+1}\big)\sin^2\big(\frac{(n+1)\pi}{2n+1}\big)}\)

0.4330127018922193233818615853764680917357,
0.5449068960040206644216000681104640622121,
0.6784585056636426221885991789493090957810,
0.8206610632070756875691817572731235996389,
0.9675499102380051550936791244123655291838,
1.117334580597582270236510738659181882934,
1.269056651101686634440984909507141812433,
1.422148171751726954319251197563916622225,
1.576248125681685807482872309444375726070,
1.731114727510611291694524417993803848020,
1.886579304584877676599013839269108328931,
2.042520253337289932329348431739277149293,
2.198847488766383717332918255753341199207,
2.355492730538745301627044238435558442138,
2.512403203438687479861576100925163259289,
2.669537420593496932808926039388713668006,
2.826862283079880127472958360606177832503,
2.984351037288439455102907912112967650801,
3.141981806238652358576477426083149027816
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 楼主| 发表于 2020-4-17 08:42:12 | 显示全部楼层
王守恩 发表于 2020-4-16 19:57
还能化简吗?

\(\D\sum_{k=1}^n\frac{\cos\big(\frac{(n-k+1)\pi}{2n+1}\big)\sin^3\big(\frac{(n+k)\ ...

31# 和 18# 公式等效,但用mathematica 不能化简
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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