By Percival Frost Joseph Wolstenholme
The Authors of the subsequent Treatise have endeavoured to offer ahead of scholars as entire a view of the topic as definite obstacles have allowed them to do. the need of those obstacles has constructed itself during getting ready the paintings in the course of a interval of 4 years. The research of innumerable papers, via the main celebrated mathematicians of all international locations, has confident the authors that the topic is nearly inexhaustible, and that, to have taken care of all elements of it with something impending to the fulness with which they've got handled the 1st component, could have swelled their paintings in a nervous percentage to what it has already attained.
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Use np = e(p ln n)/n . 8. Show that the infinite arithmetic series a + (a + d) + (a + 2d) + · · · + [a + (n − 1)d] + · · · diverges for a2 + d2 > 0. Hint. Sn = 1 2 n [2a + (n − 1)d], and lim Sn = ∞ unless a = d = 0. 9. Show that the infinite geometric series a + ar + ar 2 + · · · + ar n−1 + · · · , a = 0, a converges to if |r| < 1 and diverges if |r| ≥ 1. 1−r a − arn Hint. Sn = . 10. Use integral test to check convergence of the following series: 1 1 1 1 (a) √ + √ + √ + √ + · · · ; 3 5 7 9 1 1 1 (b) 1 + p + p + p + · · · ; 2 3 4 1 (c) sin π + 14 sin 12 π + 19 sin 13 π + 16 sin 14 π + · · · .
N an1 · · · amn xm j=1 amj xj The matrix A is also represented by A = [aij ], or by A = [c1 |c2 | · · · cj | · · · |cn ], where cj , 1 ≤ j ≤ n, denotes the jth column of A. Let A = [aij ] be an n × n matrix. If the ith row and jth column of A are deleted, the remaining (n − 1) rows and (n − 1) columns form another matrix Mij , called the minor of aij . The cofactor of [aij ] is defined as cij = (−1)i+j det[M ], such that n n i=1 aij cik = 0 for j = k, and i=1 aij ckj = 0 for i = k.
27) where L denotes the linear differential operator, and subject to the boundary conditions u(0) = 0 = u(l), which imply that the endpoints of the string are kept fixed. 27) is u(t) = A(t) cos kt + B(t) sin kt, where the functions A(t) and B(t) are to be determined by the method of variation of parameters. Thus, we solve the equations A (t) cos kt + B (t) sin kt = 0, −kA (t) sin kt + kB (t) cos kt = −f (t) to find A (t) and B (t), which are given by A (t) = f (t) sin kt , k and B (t) = − f (t) cos kt .