In this paper, we will present several algorithms for computing with D-algebraic power series. Such power series are specified by one or more algebraic differential equations and a sufficient number of initial conditions. The emphasis is not on the efficient computation of coefficients of such power series (various techniques are known for that), but rather on the ability to decide whether expressions involving D-algebraic power series are zero. We will both consider univariate and multivariate series and, besides the usual ring operations and differentiation, we will also consider composition, implicitly determined power series and monomial transformations.