Introduction A sequence transformation T is a function T : (x_k) → (x_k~*) which maps a slowly convergent sequence to another sequence with better numerical properties. If lim_k x_k = x and lim_k xk~* = x~* with r_k and rk* as the truncated errors. Then we have x_k = x + r_k, rk~* = x~* + rk~*. We say that the sequence (x_k) converge more rapidly than the sequence (xk~*) if {formula} The convergence rate of a sequence is defined as follows: Let (x_k) be a real valued sequence with limit x. Then the convergence rate of (x_k) is characterized by {formula} which closely resembles the ratio test in the theory of infinite series. If 0 < a < 1, then (x_k) is said to be linearly convergent. If a = 1, then (x_k) is said to be logarithmically convergent, if a = 0 then (x_k) is said to converge hyper-linearly and obviously a > 1 stands for divergence of the sequence.
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