Error analysis of an algorithm for summing certain finite series

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Abstract
An algorithm for summing the series , where the coefficients an are assumed known, and the quantities pn, satisfy a linear three term recurrence relation, has been given by Clenshaw [1]. If we suppose that the pn satisfy the recurrence relation where αn and βn are, in general, functions of n, then PN may be found by constructing a sequence {bn} for n = N(−1)0, where the b satisfy the inhomogeneous recurrence relation with the conditions, The sum PN is then given by This result can be readily verified by multiplying each side of equation (1.2) by pn, summing from n = 0 to N, and making use of equations (1.1) and (1.3).