Sequence Summation

Geometric Series: $a,ar,a{r}^2,\dots ,a{r}^n$

$$S_n=a\cdot \frac{1-r^n}{1-r}$$

If it converges, the infinite sum:

$$S_{\infty }=\frac{a}{1-r}$$

Arithmetic Sequence: $a,a+d,a+2d,\dots a+nd$

$$S_n=\frac{n}{2}\cdot (2a+(n-1)d)$$

Other Summations: See Convergence tests - Wikiwand

Properties of Summations §

$${\left(\sum _{i=0}^n{a}_i\right)}^2=\sum _{i=0}^n{a}_i^2+\sum _{i\ne j}^n{a}_i{a}_j$$