By R.O. Gandy, C.E.M. Yates, Solomon Feferman, Andrew Hodges, Jack Good and Martin Campbell-Kelly (Eds.)

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727-742.

W The a u t h o r has f o u n d a description of such a machine. II The n e g a t i o n sign is w r i t t e n before a n expression a n d n o t over it. 82 Asequence of r p r i m e s is d e n o t e d b y ("). ] ON COMPUTABLE NUMBERS. 253 I say t h a t a is then a computable sequence" a machine Jr to compute a can be obtained by a fairly simple modification of ~r We divide the motion of Jr into sections. The n-th section is devoted to finding the n-th figure of a. After the (n-- 1)-th section is finished a double colon :: is printed after all the symbols, and the succeeding work is done wholly on the squares to the right of this double colon.

We can then show t h a t (vii) A power series whose coefficients form a computable sequence of computable numbers is computably convergent at all computable points in the interior of its interval of convergence. (viii) The limit of a computably convergent sequence is computable. And with the obvious definition o f " uniformly computably convergent"(ix) The limit of a uniformly computably convergent computable sequence of computable functions is a computable function. Hence ( x ) The sum of a power series whose coefficients form a computable sequence is a computable function in the interior of its interval of convergence.