By Peng Wei, Sun Bingnan, Tang Jinchun

In keeping with analytical equations, a catenary aspect is gifted for thefinite point research of cable constructions. in comparison with often used aspect (3-node point, 5-node element), a application with the proposed point is of lesscomputer time and higher accuracy.

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**Example text**

6. 7. Prove that the sequence I I 2 I 3 I 4 l, 2• 3• 3• 5• 4• 7• 5• with the general term Xn = { l/n, n/(n + 2), if n=2k- 1, if n = 2k, has no limit. Solution. It is easy to show that the points x 11 with odd numbers concentrate about the point 0, and the points x,, with even numbers, about the point 1. Hence, any neighbourhood of the point 0, as well as any neighbourhood of the point 1, contains an infinite set of points x11 • Let a be an arbitrary real number. 6. Number Sequences. Limit of a Sequenc't' not contain at least a certain neighbourhood of either point 0 or point l.

Hence, the sequence converges, its limit being V2. (e) The sequence decreases monotonically. Indeed, Xn+i = Xn+l nn Since (n+ l)n (n+ 1)1 (n+ J)n+1 n! (n+ J)n < 1, Xn+l < Xn. > 0, the sequence Then, since Xn n! nn = nn • (n+ J)n nn = (n+ l)n Xn· is bounded below, hence lim xn exists. Let us denote it l. Obviously, l = Jim xn ~ 0. Now let us show that l = 0. Indeed, (n~})n =(n~I n- r a;, =(l+ ~r~ 1-tn ~ =2. 54 Ch. I. Introduction to Mathl'matical Analysis nn Hence, (n+ l)n we obtain < 2I I <2 and Xn+i xn.

I. 2. Knowing the first several terms of the sequence, write one of the possible expressions for the general term: 2 5 10 17 26 (a) 3' 8' T3' 18' 23; 1 1 I 1 (b) 1, 2' 2, 3' 3, 4' 4, 5· Note. A knowledge of the first several terms of a sequence is not sufficient to define this sequence. That is why this problem should be understood as one of finding a certain simple inductive regularity compatible with the given terms. Solution. e. n 2 1, while the denominators form the arithmetic progression 3, 8, 13, 18, ...