By Richard Reti
Richard Reti was once an impressive over the board grandmaster as proven through his victories on the greatest tournaments of Kaschau 1918, Gothenburg 1920 and Teplitz Schonau 1922. His sufferers integrated Capablanca, Alekhine and Nimzowitsch, whereas his stylish destruction of Bogolyubov deservedly gained the sweetness prize at manhattan 1924.
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Case 3. When r exceeds both ft and a but not their sum (r > ft, r > a, r < ft a), we . = can put r * Since (a -f b (r . + = (6 + c), in which c (b Hence . + - c) c) > r and r 0, so that r > > c. a, (r -f b * and < r so that + a)<'> = (r + > c) > r, c)<'>. and since r > b t so that (2r c) is also greater than r. e. : If r by = (b + a) so that c = 0, there + 3) above. Case When 4. 1 r exceeds the sum Sw = 0. of b and a (r EXERCISE 1. 06 + 5)<>. + 39)< 13 residual r. only one residual term, viz. e.
05 Ay. = r(b + a)"' = r(r - 1 ', a)"-". Hence we have A^ = r <*>i<'-" * Vandermonde's Theorem. ; ya = for integral values of b + 2)<", . = 0, . and a is Consider the series whose general )">. )">, and . 1,2, etc. 1; and *< + 2) #(# 1)(* 2) *(* 1)(* and/or a. 0=0. In general #< r) 0, if r > x. Hence (b a) (r) vanishes if r > (a ft) (r) Vanish if the index (x) of and all terms in the power series corresponding to (6 a) - = - - = . = + + + a exceeds of b exceeds a, or that Consider the expansion b.
1 (1 (ii) + a)n ~ *. values of 5 w2 T* 10. 10 We - * 1 - rr-i l + X 2-5 : ~' (iii) ; | log. ; (iv) tan-* *. FINITE INTEGRATION can regard integration as the solution of a differential equation inasmuch as the evaluation left hand expression below is equivalent to finding the unknown of the expression to A of the the right of it : AA = f \y- dd j dx > Tx =y ...... 0) We we may speak of integration as the operation inverse to differentiation. e. : Accordingly, now see Thus the figurate number of dimension d and rank n in Fig.