text: The case of recurrent sequences  which occurs in dynamical systems and in the context of various fixed-point theorems is of particular interest. Assuming that the relevant derivatives of f are continuous, one can (easily) show that for a fixed point  such that , one has at least linear convergence for any starting value  sufficiently close to p. If   and  , then one has at least quadratic convergence, and so on. If , then one has a repulsive fixed point and no starting value will produce a sequence converging to p (unless one directly jumps to the point p itself).
  Acceleration of convergence
  Many methods exist to increase the rate of convergence of a given sequence,
  i.e. to transform a given sequence into one converging faster to the same limit.  Such techniques are in general known as "series acceleration". The goal of the transformed sequence is to reduce the computational cost of the calculation. One example of series acceleration is Aitken's delta-squared process. These methods in general (and in particular Aitken's method) do not increase the order of convergence, and are useful only if initially the convergence is not faster than linear: If  convergences linearly, one gets a sequence  that still converges linearly (except for pathologically designed special cases), but faster in the sense that . On the other hand, if the convergence is already of order ≥ 2, Aitken's method will bring no improvement.