Increasing the convergence rate of series

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Abstract

This paper deals with the question of obtaining from the sequence {sn} of partial sums of a convergent series s a new sequence {tn} which converges to the same limit s as sn, but more rapidly. When the general term un of the series s possesses certain types of expansion involving inverse powers of n, it is shown how tn is obtained by adding a fixed number of terms to sn. When the series s is convergent, these terms tend to zero as n tends to infinity, but they are such as to make tn much more rapidly convergent to s—in fact we can make the convergence rate as great as we wish. Explicit general formulas are obtained for a wide range of important series.

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