# How to prove this statement of the geometric series

If we have $$sum_ {n = 0} ^ { infty} r ^ {i}$$, that converges towards $$S = frac {1} {1-r}$$, shows CA $$S – s_n = frac {r ^ {n + 1}} {1-r}$$, or $$s_n$$ is the sum of the first n terms.

My approach,

$$S – s_n = frac {1} {1-r} – frac {1-r ^ n} {1-r} = frac {r ^ n} {1-r}$$

So, there is no $$r ^ {n + 1}$$ in my answer as in the question. If we have $$s_n$$, that means the sum of the first $$n$$ terms, so there should not be any $$n + 1$$.

Where am I wrong?