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Continued Fractions of Tails of Hypergeometric Series

1. MOTIVATION. The tails of the Taylor series for many standard functions such as the arctangent and the logarithm can be expressed as continued fractions in a variety of ways. A surprising side effect is that some of these continued fractions provide a dramatic acceleration for the convergence of the underlying power series. These investigations were motivated by a surprising observation about Gregory’s series. Gregory’s series for π, truncated at 500,000 terms, gives to forty places