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By Jon Speelman

Unique discoveries within the endgame unearthed via Britain's top specialist and global championship challenger.

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When x by evaluating the arbitrary constants we write as QJ which = 0, f(x) = A we have =A + l write this as briefly /^O) = AI and in general for the value of the nth derivative we write fn (x), whose = 0. A + f*(x) 2 a 3 . . t . . t . 4 t * 47 z . . . . Hence we may put o\ 2t\ By substitution of these values in the series for y v2 When j = log. (1 4! +*) =+2(1 ' +*)-'. +*) = -3. 2. +*)= +4. 3. When * 1 2. (1 +*)-. = 0, ^ = log, 1 = 0, so that/(0) = 0. , 1, 1, so that x = than unity, this must be convergent.

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. 5 6 (2+3)< >=5< >=5! By (i) (2 above + 3) = 2< + 5 (5) 6) . + 2 (4) 3 10 . 2 (3) 3< 2) + 10 . 2. 3< 4 + 3< > 5) . 2 (2) 3 (3) the index of 2 exceeds 2 or the index of 3 exceeds 3. Accordingly, all terms vanish except , 10 The number is of powers, examination.

For the multinomial expansion, the corresponding expression If (1 +1+ 1 . . e. )' 1 (1 ' ^ul 1 basic terms, = m'. Vll ... vital EXERCISE Expand 2. Give the terms of the expansion (i 3. Find the numerical values of successive terms of : +i+ (* Compare the results of 4 *) - + and and (| (| + i + i)*. + i + J)t. 08 a, 6, r, . +i+ (*)(}+' 5. 07 1. 4. is +i+ 4 i) ; (* + & + #. , are all positive integers it is : FACTORIAL POWERS IN possible to + + + + expand (a +b+c+d (r) . ) r

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