1 The Various Kinds of Algorithmic Sequence Transformations -- 1 โ{128}{148} Sequence Transformations -- 2 โ{128}{148} Algorithms for Sequences and Algorithmic Transformations -- 3 โ{128}{148} k-Normal Algorithms and k-Normal Transformations -- 4 โ{128}{148} k-Memories Algorithms and k-Memories Transformations -- 5 โ{128}{148} k-Stationary Algorithms and k-Stationary Transformations -- 6 โ{128}{148} Rational Transformations and Linear Transformations -- 7 โ{128}{148} Diagram of Inclusions -- References -- 2 Decidability and Undecidability in the Limit -- 1 โ{128}{148} Definitions and the Normalisation Theorem -- 2 โ{128}{148} Problems Concerning Convergence, Turbulence and Periodicity of Sequences -- 3 โ{128}{148} Algorithms for Counting the Number of Accumulation Points -- 4 โ{128}{148} Algorithms for Determining the Period of an Asymptotically Periodic Sequence -- 5 โ{128}{148} Families of Sequences of Iterations -- 6 โ{128}{148} Two General Results Concerning the Decidability in the Limit -- Appendix 1 โ{128}{148} Strength of an Accumulation Point and Quickness of a Sequence -- Appendix 2 โ{128}{148} Decidability in the Limit and Recursivity. -- Appendix 3 โ{128}{148} Decidability of the Convergence, Turbulence and Asymptotic Periodicity of a Continuous Function -- References -- 3 Algorithms for Extracting Convergent Subsequences -- 1 โ{128}{148} T-Algorithms -- 2 โ{128}{148} S-Algorithms -- 3 โ{128}{148} U-Algorithms -- 4 โ{128}{148} Limitation Results -- References -- 4 The Partially Ordered Systems of Accelerable Families -- 1 โ{128}{148} Acceleration Velocity, Acceleration, Prediction -- 2 โ{128}{148} Transformations for Convergence Acceleration, Accelerable Families -- 3 โ{128}{148} Examples of Accelerable Families of Sequences -- 4 โ{128}{148} Relationships Between the Ordered Systems of Accelerable Families -- 5 โ{128}{148} Maximal Accelerable Families -- References -- 5 Non-Accelerable Families of Sequences -- 1 โ{128}{148} Remanence and First Applications -- 2 โ{128}{148} Families of Monotones Sequences -- 3 โ{128}{148} Alternating and Oscillating Sequences -- 4 โ{128}{148} Families of Linearly Convergent Sequences -- 5 โ{128}{148} Families of Logarithmically Convergent Sequences -- 6 โ{128}{148} Table of Results -- References -- 6 Accelerating the Convergence of Linear Sequences -- 1 โ{128}{148} Linearly Convergent and Periodico-Linearly Convergent Sequences -- 2 โ{128}{148} Acceleration of Periodico-Linear Sequences -- 3 โ{128}{148} Optimality of the ?2 of Aitken -- References -- 7 Automatic Selection of Sequence Transformations -- 1 โ{128}{148} General Methods -- 2 โ{128}{148} Automatic Choice of Sequences of Parameters in the Richardson Extrapolation -- References