Somatório e Produtório

Propriedades de Somatório
$$ \sum_{n=s}^t C\cdot f(n) = C\cdot \sum_{n=s}^t f(n)\text{, onde C é uma constante. $$

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

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

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

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

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

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

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

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

$$ \sum_{i=s}^m\sum_{j=t}^n {a_i}{c_j} = \sum_{i=s}^m a_i \cdot \sum_{j=t}^n c_j $$

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

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

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