By H W Turnbull

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For a proof we may refer to We may regard the statement a just geometrical progression summed to infinity or equally well as an example of the binomial l 1 1 expansion of ax~ (l aor )"" Now take \x\ greater than the greatest among |a|, each fraction, and add the results. , |/3 |A|, expand any text on analysis. written as . , (2) the development of R(x] in descending powers of and for valid x, sufficiently large values of \x\. ). , |A|, E(x) is = f(x)/(x) = -Zaa- Unless one or other of a, Therefore if # possible.

II(xa) r quadratic case, when

Show from the zigzag that 2x* has a root between 16. and + 8, = x -f 1 1. , digit by digit we need a practical way of expressing /(x) as a polynomial g(y) where y x+a. Equation. +b n _ y+b n g(y) be identically equal when y = xa. Thmf(x) and to are efficients Q l l Q 1 l a t are supposed known to find the coefficients b t ; so too is a. . (I) . (2) The We co- require . , (3) which exhibits/(x) as a polynomial in y of degree n. Hence b n =/(a), 6 n _! = f'(a) and so on. Incidentally this shows that the degree n is the same both for g(y) in y and f(x) in x.