Summation Formulas

Geometric series formula

$$\sum _{k=0}^{n}a{r}^k=\frac{a(1-r^{n+1})}{1-r}$$

The following can be derived from the above equation:

When $|r|<1$, then as $n$ approaches $\infty$,

$$\sum _{k=0}^{\infty }a{r}^k=\frac{a}{1-r}$$

When $r=1$, then

$$\sum _{k=0}^{n}a=a(n+1)$$