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Abel's Inequality Abel's Theorem absolutely convergent adoo algebraic function algebraical apply ascending powers Binomial Series Binomial Theorem calculate Cauchy Cauchy's chap coefficient commensurable condition continued fraction convergent or divergent convergent series corresponding cosh deduce definite denote divergent according double series equation Euler Example EXERCISES expansion exponential factors finite number finite value follows at once formula geometric geometric series given graph Hence hyperbolic functions hypergeometric series indeterminate form inequality infinite products infinite series integral function integral solutions integral values less letters limit logarithmic means multiply number of terms obvious oscillates permutations positive integer positive or negative positive quantities prime prove ratio residue result scries semi-convergent simple continued fraction sinh sufficient sum the series sum to infinity summation suppose theory tion ultimately vanish vergent xxvi xxvii zero
Page 33 - There are n points in a plane, no three of which are in the same straight line, with the exception of p, which are all in the same straight line ; find the number of lines which result from joining them.
Page 538 - If on taking any very large number N out of a series of cases in which an event A is in question, A happens on pN occasions, the probability of the event A is said to be p.
Page 71 - The limit can be found by dividing numerator and denominator by the highest power of x that occurs in either.
Page vii - Every mathematical book that is worth reading must be read "backwards and forwards," if I may use the expression. I would modify Lagrange's advice a little and say, "Go on, but often return to strengthen your faith." When you come on a hard or dreary passage, pass it over; and come back to it after you have seen its importance or found the need for it further on.
Page 48 - ... power of their arithmetical mean, according as m does not or does lie between 0 and 1.
Page 563 - A and B each take twelve counters and play with three dice on this condition, that if eleven is thrown, A gives a counter to B, and if fourteen is thrown, B gives a counter to A; and he wins the game who first obtains all the counters. Show that A's chance is to B's as 244 14o625 is to 282429536481.
Page 12 - A man has 7 relatives, 4 of them are ladies and 3 gentlemen, his wife has 7 relatives and 3 of them are ladies and 4 gentlemen. In how many ways can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man's relatives and 3 of wife's relatives ? Solution.
Page 94 - ... is the tangent of the inclination to the axis of x of the...
Page 506 - If the equation rj? + y3 = ?i had an integral solution, show that one of the three x, y, z must be of the form 7m, and one of the form 3m. (36.) The area of a right-angled triangle with commensurable sides cannot be a square number. (37.) The sum of two integral fourth powers cannot be an integral square.