How do you know what $\sum_{n=a}^{\infty}p^n$ converges to if you know $\sum_{n=0}^{\infty}p^n = \frac{1}{1-p}$? How do you know what $\sum_{n=a}^{\infty}p^n$ if you know  $\sum_{n=0}^{\infty}p^n = \frac{1}{1-p}$ ?
Apparently $\sum_{n=a}^{\infty}p^n$ = $\frac{p^a}{1-p}$, but how can you derive this? 

Because of the high-school identity:
$$1-p^n=(1-p)(1+p+p^2+\dots+p^{n-1}),$$
rewritten as 
$$\frac1{1-p}=1+p+p^2+\dots+p^{n-1}+\frac{p^n}{1-p}, $$
which shows that
$$\Biggl\lvert\frac1{1-p}-(1+p+p^2+\dots+p^{n-1})\Biggr\rvert=\frac{\lvert p^n\rvert}{\rvert 1-p\rvert},$$
and the latter fraction tends to $0$ if $\lvert p\rvert<1$.
