深度学习基础 - 累加符号和连乘符号

深度学习基础 - 累加符号和连乘符号

flyfish

累加符号 其他名字 Sigma Notation 、Summation Notation

示例1

$\begin{array}{rl}& \sum _{i=1}^{5}i\\ & =1+2+3+4+5\end{array}$

示例2

$\begin{array}{rl}& \sum _{i=3}^{7}i\\ & =3+4+5+6+7\end{array}$

示例3

$\begin{array}{rl}& \sum _{i=2}^{5}2i\\ & =2(2)+2(3)+2(4)+2(5)\\ & =4+6+8+10\end{array}$

示例4

$\begin{array}{rl}& \sum _{j=1}^{4}jx\\ & =1x+2x+3x+4x\end{array}$

示例5

$\begin{array}{rl}& \sum _{i=1}^2\sum _{j=4}^6(3ij)\\ & =\sum _{i=1}^2(3i\cdot 4+3i\cdot 5+3i\cdot 6)\\ & =(3\cdot 1\cdot 4+3\cdot 1\cdot 5+3\cdot 1\cdot 6)+(3\cdot 2\cdot 4+3\cdot 2\cdot 5+3\cdot 2\cdot 6)\end{array}$

示例6

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

连乘符号 其他名字 Pi Notation、Product Notation

示例1

$\begin{array}{rl}& \prod _{i=3}^{7}i\\ & =(3)(4)(5)(6)(7)\end{array}$

示例2

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

示例3

$\begin{array}{rl}& \prod _{i=1}^2\prod _{j=4}^6(3ij)\\ & =\prod _{i=1}^2((3i\cdot 4)(3i\cdot 5)(3i\cdot 6))\\ & =((3\cdot 1\cdot 4)(3\cdot 1\cdot 5)(3\cdot 1\cdot 6))((3\cdot 2\cdot 4)(3\cdot 2\cdot 5)(3\cdot 2\cdot 6))\end{array}$