深度學習基礎 - 累加符號和連乘符號

深度學習基礎 - 累加符號和連乘符號

flyfish

累加符號 其他名字 Sigma Notation 、Summation Notation

示例1

$\displaystyle \begin{aligned} &\sum_{i=1}^{5} i\\ &=1+2+3+4+5 \end{aligned}$

示例2

$\displaystyle \begin{aligned} &\sum_{i=3}^{7} i\\ &=3+4+5+6+7 \end{aligned}$

示例3

$\displaystyle \begin{aligned} &\sum_{i=2}^{5} 2 i\\ &=2(2)+2(3)+2(4)+2(5)\\ &=4+6+8+10 \end{aligned}$

示例4

$\displaystyle \begin{aligned} &\sum_{j=1}^{4} j x\\ &=1 x+2 x+3 x+4 x \end{aligned}$

示例5

$\displaystyle \begin{aligned} &\sum_{i=1}^{2} \sum_{j=4}^{6}(3 i j)\\ &=\sum_{i=1}^{2}(3 i \cdot 4+3 i \cdot 5+3 i \cdot 6)\\ &=(3 \cdot 1 \cdot 4+3 \cdot 1 \cdot 5+3 \cdot 1 \cdot 6)+(3 \cdot 2 \cdot 4+3 \cdot 2 \cdot 5+3 \cdot 2 \cdot 6) \end{aligned}$

示例6

$\displaystyle \sum_{k=1}^{\infty}(k+1)^{3}=2^{3}+3^{3}+\cdots n^{3}+\cdots$

連乘符號 其他名字 Pi Notation、Product Notation

示例1

$\displaystyle \begin{aligned} &\prod_{i=3}^{7} i\\ &=(3)(4)(5)(6)(7) \end{aligned}$

示例2

$\displaystyle \prod_{i=1}^{6} i^{2}=(1)(4)(9)(16)(25)(36)$

示例3

$\displaystyle \begin{aligned} &\prod_{i=1}^{2} \prod_{j=4}^{6}(3 i j)\\ &=\prod_{i=1}^{2}((3 i \cdot 4)(3 i \cdot 5)(3 i \cdot 6))\\ &=((3 \cdot 1 \cdot 4)(3 \cdot 1 \cdot 5)(3 \cdot 1 \cdot 6))((3 \cdot 2 \cdot 4)(3 \cdot 2 \cdot 5)(3 \cdot 2 \cdot 6)) \end{aligned}$