T o t a l = ∑ n = 1 ⌊ 1 4 ∑ n = 1 n − 1 ⌊ n + 300 × 2 n / 7 ⌋ ⌋ − 1 ( n 4 + 75 × 2 n / 7 ) = L ( L − 1 ) 8 + 75 ⋅ ( 2 L 7 − 2 1 7 2 1 7 − 1 ) {\displaystyle Total = \sum_{n = 1} ^ {\left \lfloor \frac{1}{4}\displaystyle\sum_{n = 1}^{n-1} \left\lfloor n + 300 \times 2^{n/7} \right\rfloor \right\rfloor - 1} \left( \frac {n} {4} + 75 \times 2 ^ {n/7} \right)=\frac{L(L-1)}{8}+75\cdot \left(\frac{2^{\frac{L}{7}}-2^{\frac{1}{7}}}{2^{\frac{1}{7}}-1} \right) }
π = 12 ∑ k = 0 ∞ ( − 3 ) − k 2 k + 1 = 12 ∑ k = 0 ∞ ( − 1 3 ) k 2 k + 1 = 12 ( 1 − 1 3 ⋅ 3 + 1 5 ⋅ 3 2 − 1 7 ⋅ 3 3 + ⋯ ) {\displaystyle \pi = \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1} = \sqrt{12}\sum^\infty_{k=0} \frac{(-\frac{1}{3})^k}{2k+1} = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)}
