An extension of the Levin-Sidi class of nonlinear transformations for accelerating convergence of infinite integrals and series (Q1262688)

The paper deals firstly with the evaluation of integrals \(S=\lim F(x)\quad (x\to \infty)\) where \(F(x)=\int^{x}_{0}f(t)dt.\) It is supposed that \(\epsilon (x)=S-F(x)\) satisfies a linear differential equation, the coefficients of the derivatives of \(\epsilon\) being series in inverse powers of x with constant coefficients. If it is assumed that all series concern terminate, i.e. that they are polynomials in the variable 1/x, the constant polynomial coefficients may be eliminated from expressions involving \(F\{\) x(0)\(\}\),...,F\(\{\) x(n)\(\}\) where n is suitably chosen, to yield a determinantal quotient D for S involving the F values together with values of f and its derivatives at x(0),...,x(n). In the absence of an assumption concerning the polynomial nature of the coefficients in the differential equation satisfied by \(\epsilon\), D offers an approximation to S. The above treatment is extended to sums \(S=\lim F(x)\quad (x\to \infty)\) where \(F(x)=\sum f(t)\quad (0\leq t\leq x),\) x and t being integers and \(\epsilon\) now satisfying a linear difference equation. The use of the quotient D is, as are all transformations involving the direct use of determinants, vitiated in practice by formidable numerical instability.