# Messenger of Mathematics Volume 4

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1868 edition. Excerpt: ...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 (19-i), Vol. III., p. 256, line 5, dele the first two terms of the coefficient of Y. "Oakwal" Near Brisbane, Queensland, Australia, January 22, 1867. a + cf + a +...+ ar 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"12-! n + 1 aa'n-l) n W or if a-1 = b, since a2n+2 = (b + l)2n+2, that hTM + (2n + 2) VTM + (2w+2)J2 + 1) bm + &c. + (2m + 2) b Now on the left hand there are (2n + 2) terms in the numerator, and 2n terms in the denominator. Let us consider the r1h term from the end in the numerator and the r4h term from the end in the denominator. (1) may then be written nbn + 2nn+l)Vn+1 + &c. where r has positive integral values from 1 to 2w. Now the coefficient of V will be positive, if a has such a value that (2w + l)(2w + 2) w4 1 wX (2n + l-r)(2n + 2-r) a ' 0r lf a 2w(2- + 1) (3)' or a fortiori, if a c, when c is the greatest value which the right-hand side of (3) can have. Now the numerator of (3) = r-r in + 3) + (2w + 1)(2m + 2) = (r-2n-l-)'-i; therefore the right-hand side of (3) will be as large as possible, when r is as small as possible, i.e. when it = 1, and then 2m (2ra +1) Therefore, if a 1, the coefficients of all the powers of b in (2) are positive. But a 1 is the hypothesis with which we started, therefore the thing is proved in this case. Next, if al, ...show more

## Product details

• Paperback | 50 pages
• 189 x 246 x 3mm | 109g
• Miami Fl, United States
• English
• black & white illustrations
• 1236577809
• 9781236577801