This nal project discusses an error sums from roots of positive integers but not a perfect square. The error sum functions dened by "( √ D) = Σ1 m=0(−1)m(qm √ D − pm) and "( √ D) = Σ1 m=0(qm √ D − pm), are considered where D is positive integers but not a perfect square and pm=qm denotes the convergence of the continued fraction expansion of √ D. This is …