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?