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1 The Various Kinds of Algorithmic Sequence Transformations.- 1 - Sequence Transformations.- 2 - Algorithms for Sequences and Algorithmic Transformations.- 3 - k-Normal Algorithms and k-Normal Transformations.- 4 - k-Memories Algorithms and k-Memories Transformations.- 5 - k-Stationary Algorithms and k-Stationary Transformations.- 6 - Rational Transformations and Linear Transformations.- 7 - Diagram of Inclusions.- References.- 2 Decidability and Undecidability in the Limit.- 1 - Definitions and the Normalisation Theorem.- 2 - Problems Concerning Convergence, Turbulence and Periodicity of Sequences.- 3 - Algorithms for Counting the Number of Accumulation Points.- 4 - Algorithms for Determining the Period of an Asymptotically Periodic Sequence.- (a) Methods of Detector Coefficients.- (b) Methods Using Barycenters.- (c) Limitation Results.- 5 - Families of Sequences of Iterations.- 6 - Two General Results Concerning the Decidability in the Limit.- Appendix 1 - Strength of an Accumulation Point and Quickness of a Sequence.- (a) Strength of an Accumulation Point.- (b) Quickness of a Sequence.- Appendix 2 - Decidability in the Limit and Recursivity..- Appendix 3 - Decidability of the Convergence, Turbulence and Asymptotic Periodicity of a Continuous Function.- References.- 3 Algorithms for Extracting Convergent Subsequences.- 1 - T-Algorithms.- 2 - S-Algorithms.- 3 - U-Algorithms.- 4 - Limitation Results.- References.- 4 The Partially Ordered Systems of Accelerable Families.- 1 - Acceleration Velocity, Acceleration, Prediction.- 2 - Transformations for Convergence Acceleration, Accelerable Families.- 3 - Examples of Accelerable Families of Sequences.- 4 - Relationships Between the Ordered Systems of Accelerable Families.- 5 - Maximal Accelerable Families.- References.- 5 Non-Accelerable Families of Sequences.- 1 - Remanence and First Applications.- 2 - Families of Monotones Sequences.- 3 - Alternating and Oscillating Sequences.- 4 - Families of Linearly Convergent Sequences.- 5 - Families of Logarithmically Convergent Sequences.- 6 - Table of Results.- References.- 6 Accelerating the Convergence of Linear Sequences.- 1 - Linearly Convergent and Periodico-Linearly Convergent Sequences.- (a) Problems on Convergence.- (b) Problems on Pseudo-Periodicity.- (c) Applications.- 2 - Acceleration of Periodico-Linear Sequences.- 3 - Optimality of the ?2 of Aitken.- (a) Algebraical Optimality.- (b) The Impossibility of Enlarging Lin.- (c) Acceleration of Degree 1+s on Lin.- References.- 7 Automatic Selection of Sequence Transformations.- 1 - General Methods.- (a) General Definitions for Selection Methods.- (b) Two Results on Selection Methods.- (c) Numerical Experiments.- 2 - Automatic Choice of Sequences of Parameters in the Richardson Extrapolation.- (a) Exactness Properties of the Transformations Obtained from the Richardson Process.- (b) Selection Among k-th Column Transformations.- (c) Selection Among k-th Diagonal Transformations.- (d) Selection Among Fast Diagonal Transformations.- References.