The Oxford, Cambridge, and Dublin Messenger of Mathematics Volume 4 (Paperback)

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1868 edition. Excerpt: ...only differing infinitesimally from 1. In the former case 1--1 ] &c. has its terms, all, identical in value though differing in sign. In the latter, I--1 + &c. only represents the beginning of the series, while the infinitely distant terms are not +1. I conclude by stating that, though I do not positively assert that the mean value of the series is not the true value, it is conceivable that in some cases the data of a question may enable us to determine the ratio n Theorem, &o. p. 2, of separate copy. as an even or an odd integer or both; and that in such case the expression must be employed to evaluate the series. P.S. In equation (194), Vol. in., p. 256, line 5, dele the first two terms of. the coefficient of Y. " Oakwal" Near Brisbane, Queensland, Australia, January 22, 1867. AN INEQUALITY. By H. J. Sharpe, Fellow of St. John's College, Cambridge. To prove that if a 1, l+a? + ai+...+ a"' n + 1 a + a2 + a6+...+ a2"1 n Two articles on this inequality have already appeared in the Messenger, one by Dr. Ingleby on p. 235, Vol. III., and one on p. 39, Vol. IV. The first article is very ingenious, but decidedly artificial. In the last article it is difficult to see whether anything is proved or not. I will attempt to give another proof. First, suppose a 1, then we have to prove that a--I n + 1, . Mi Now on the left hand there are (2m+ 2) terms in the numerator, and 2n terms in the denominator. Let us consider the r1b term from the end in the numerator and the r1u term from the end in the denominator. (1) may then be written nbta + 2n (n+l)J'"+1 + &c. where d 1, but this was proved in the first case. We may give another proof of (a) by the Theory of Equations. In it write x for a. Consider the equation y...