如何計算：
lim n → ∞ ∑ j = 0 n G 2 n sin j π n {\displaystyle \lim _{n\to \infty }\sum _{j=0}^{n}{\frac {G}{2n}}\sin {\frac {j\pi }{n}}} （G為常數）
能用積分表示就行。
（具體的數學語法參見數學公式幫助）
lim n → ∞ ∑ j = 0 n G 2 n sin j π n = lim n → ∞ [ G 2 n × ∑ j = 0 n sin j π n ] = lim n → ∞ [ G 2 n × ∫ 0 n sin j π n d j ] = lim n → ∞ [ G 2 n × ∫ 0 π sin j π n d ( j π n ) × n π ] = lim n → ∞ [ G 2 n × ∫ 0 π sin x d x × n π ] ( x = j π n ) = lim n → ∞ [ G 2 n × 2 × n π ] = G π {\displaystyle {\begin{aligned}&{\underset {n\to \infty }{\mathop {\lim } }}\,\sum \limits _{j=0}^{n}{\frac {G}{2n}}\sin {\frac {j\pi }{n}}\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \sum \limits _{j=0}^{n}{\sin {\frac {j\pi }{n}}}\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{n}{\sin {\frac {j\pi }{n}}}\operatorname {d} j\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{\pi }{\sin {\frac {j\pi }{n}}\operatorname {d} \left({\frac {j\pi }{n}}\right)\times {\frac {n}{\pi }}}\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{\pi }{\sin x\operatorname {d} x}\times {\frac {n}{\pi }}\right]\quad \left(x={\frac {j\pi }{n}}\right)\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times 2\times {\frac {n}{\pi }}\right]\\&={\frac {G}{\pi }}\\\end{aligned}}}
--Bcnof留言 2009年11月5日 (四) 10:03 (UTC)
lim n → ∞ ∑ j = 0 n G 2 n sin j π n = lim n → ∞ [ G 2 π × ∑ j = 0 n sin j π n × π n ] = G 2 π × ∫ 0 π sin x d x = G 2 π × 2 = G π {\displaystyle {\begin{aligned}&{\underset {n\to \infty }{\mathop {\lim } }}\,\sum \limits _{j=0}^{n}{\frac {G}{2n}}\sin {\frac {j\pi }{n}}\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2\pi }}\times \sum \limits _{j=0}^{n}{\sin {\frac {j\pi }{n}}\times {\frac {\pi }{n}}}\right]\\&={\frac {G}{2\pi }}\times \int _{0}^{\pi }{\sin x\operatorname {d} x}\\&={\frac {G}{2\pi }}\times 2\ \\&={\frac {G}{\pi }}\\\end{aligned}}}
--Xnj920327 (留言) 2009年11月27日 (五) 13:42 (UTC)
（G為常數）
![{\displaystyle {\begin{aligned}&{\underset {n\to \infty }{\mathop {\lim } }}\,\sum \limits _{j=0}^{n}{\frac {G}{2n}}\sin {\frac {j\pi }{n}}\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \sum \limits _{j=0}^{n}{\sin {\frac {j\pi }{n}}}\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{n}{\sin {\frac {j\pi }{n}}}\operatorname {d} j\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{\pi }{\sin {\frac {j\pi }{n}}\operatorname {d} \left({\frac {j\pi }{n}}\right)\times {\frac {n}{\pi }}}\right]\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times \int _{0}^{\pi }{\sin x\operatorname {d} x}\times {\frac {n}{\pi }}\right]\quad \left(x={\frac {j\pi }{n}}\right)\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2n}}\times 2\times {\frac {n}{\pi }}\right]\\&={\frac {G}{\pi }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2175b463ed70148ee92ff3766dacfb24f14ee3ff)
![{\displaystyle {\begin{aligned}&{\underset {n\to \infty }{\mathop {\lim } }}\,\sum \limits _{j=0}^{n}{\frac {G}{2n}}\sin {\frac {j\pi }{n}}\\&={\underset {n\to \infty }{\mathop {\lim } }}\,\left[{\frac {G}{2\pi }}\times \sum \limits _{j=0}^{n}{\sin {\frac {j\pi }{n}}\times {\frac {\pi }{n}}}\right]\\&={\frac {G}{2\pi }}\times \int _{0}^{\pi }{\sin x\operatorname {d} x}\\&={\frac {G}{2\pi }}\times 2\ \\&={\frac {G}{\pi }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5528195a84f74bc6aa082dfa3e1b32244e4ecbfb)
