Somatório e Produtório

Propriedades de Somatório

$\sum _{n=s}^{t}C\cdot f(n)=C\cdot \sum _{n=s}^{t}f(n)$ , onde C é uma constante.

$\sum _{n=s}^{t}f(n)+\sum _{n=s}^{t}g(n)=\sum _{n=s}^{t}\left[f(n)+g(n)\right]$

$\sum _{n=s}^{t}f(n)-\sum _{n=s}^{t}g(n)=\sum _{n=s}^{t}\left[f(n)-g(n)\right]$

$\sum _{i=m}^{n}f(i)=\sum _{i=m+p}^{n+p}f(i-p)$

$\sum \limits _{n=s}^{t}j=\sum \limits _{n=1}^{t}j-\sum \limits _{n=1}^{s-1}j$

$\sum _{n=s}^{j}f(n)+\sum _{n=j+1}^{t}f(n)=\sum _{n=s}^{t}f(n)$ , note que $s\leq j\leq t$

$\sum _{i=m}^{n}i={\frac {n(n+1)}{2}}-{\frac {m(m-1)}{2}}={\frac {(n+1-m)(n+m)}{2}},$ progressão aritmética.

$\sum _{i=0}^{n}i=\sum _{i=1}^{n}i={\frac {n(n+1)}{2}}$

$\sum \limits _{k=0}^{n-1}{2^{k}}=2^{n}-1$

$\sum _{i=s}^{m}\sum _{j=t}^{n}{a_{i}}{c_{j}}=\sum _{i=s}^{m}a_{i}\cdot \sum _{j=t}^{n}c_{j}$

$\sum _{i=0}^{n}i^{3}=\left(\sum _{i=0}^{n}i\right)^{2}$

$\sum _{i=m}^{n-1}a^{i}={\frac {a^{m}-a^{n}}{1-a}}(m<n)$

$\sum _{i=0}^{n-1}a^{i}={\frac {1-a^{n}}{1-a}}$

$\sum _{i=1}^{n}x_{i}=x_{1}+x_{2}+...+x_{n}$

$\sum _{i=1}^{n}x_{i}^{2}=x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}$

$(\sum _{i=1}^{n}x_{i})^{2}=(x_{1}+x_{2}+...+x_{n})^{2}$

$\sum _{i=1}^{n}x_{i}*y_{i}=x_{1}*y_{1}+x_{2}*y_{2}+...+x_{n}*y_{n}$

$(\sum _{i=1}^{n}x_{i})*(\sum _{j=1}^{m}y_{j})=(x_{1}+x_{2}+...+x_{n})*(y_{1}+y_{2}+...+y_{n})$