By John Todd

Numerical research.

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2. R e p e a t Problem 1 in the case when t h e inequality is (η + 1)-^ + 2(ηχ + 1 ) - ^ < ε . 3. 4. Suppose that M(X) = X M^(X), t h e series being convergent in t h e interval [ 0 , 1 ] and that u(x) = MO(^) + U I ( X ) + . . + u „ _ i ( x ) + r , ( x ) . L e t = maxo

Uniform Convergence and Approximations Definition 4 . The Bernstein polynomials continuous in [ 0 , 1 ] are defined by Theorem 5. B„(/, x ) - ^ / ( x ) uniformly B^{f,x) 59 for a function fix) in [ 0 , 1 ] . T a k e the case / ( x ) = x^. T h e n an easy calculation with binomial coeffi cients (cf. (/,x)-x^|<(4n)-^ for all X in [ 0 , 1 ] ; hence the convergence of B„ to x^ is uniform, and we have verified t h e t h e o r e m in this special case. A proof of the t h e o r e m in t h e general case can b e motivated t h r o u g h elementary probabilistic considerations.

It can be written as which shows that convergence is linear, N being between 0 and 1. Observe that in this case we get convergence to N- I no matter how is chosen. Consider next the sequence defined by Xo (2) This is illustrated graphically in Fig. 2. It is clear geometrically that if I Xo = 0 or X o = 2N- then Xl = X2 = ... = 0 so that there is convergence to O. Further, if 0 < X o < N- I then x" increases steadily to N- I ; if Xo = N- I then x" =N- I and if N- I