Acceleration methods based on convergence tests (Q751161)

The following simple theorem offers both a convergence test for a real valued sequence and a derived sequence which converges faster to the same limit. Let \(\{\) S(n)\(\}\) and \(\{\) x(n)\(\}\) be monotone, the latter converging to x. Set \(A(k,n)=\{x(n+1)-x(n)\}/\{S(n+k+1)-S(n+k)\}.\) If, with k fixed, lim A(k,n)\(\neq 0\), then \(\{\) S(n)\(\}\) converges and, setting \(T(n)=S(n)+\{x-x(n-k)\}/A(k,n-k),\lim \{S-T(n)\}/\{S-S(n)\}=0,\) where \(S=\lim S(n).\) The convergence tests of d'Alembert, Cauchy, Kummer, Raabe, Gauss and others are considered in the light of the above result.