Playing around with geometric series I got confused. First, consider the following equation
$$\sum_{i=0}^n 1^i = n+1$$
Now, replacing $1$ by $\frac{a}{a}$ gives
$$\sum_{i=0}^n \left(\frac{a}{a}\right)^i = n+1$$
Also, replacing $\frac{a}{a}$, by $r$ we have
$$\sum_{i=0}^n r^i = \frac{r^{n+1}-1}{r-1}$$
and replacing $r$ by $\frac{a}{b}$, we get
$$\sum_{i=0}^n \left(\frac{a}{b}\right)^i = \frac{\frac{a}{b}^{n+1}-1}{\frac{a}{b}-1} = \frac{a \left(\frac{a}{b}\right)^n-b}{a-b}$$
For the special case, where $a=b$,
$$\sum_{i=0}^n \left(\frac{a}{b}\right)^i =\sum_{i=0}^n \left(\frac{a}{a}\right)^i = \sum_{i=0}^n 1^i$$
and therefore,
$$n+1 = \frac{a \left(\frac{a}{b}\right)^n-b}{a-b} = \frac{a \left(\frac{a}{a}\right)^n-a}{a-a} = \frac{0}{0}$$
Where is my mistake?