By Serge Lang (auth.)

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**Example text**

The series for sin z, cos z, £I, etc. are to be viewed as formal series. 1. Give the terms of order (a) £I sin z (d) (g) 3 in the power series: (b) (sin zXcos z) 1 cos z e"-cosz (e) - z sin z cos z ~ (c) e% - 1 z (f) cos z sin z (h) e"/sin z 2. Define the Bernoulli numbers Bn by the power series _z_= £1-1 f B0z". n=On! Prove the recursion formula Bo n! O! + (n Bl - I)! I! -l + ... + I! (n - I)! = {I 1, if n = 0 if n> 1. Then Bo = 1. Compute B 1, B 2 , B 3 , B 4 , B 6 , B 8 , B 10 , B 12 , B 14 • Show that Bn = 0 if n is odd # 1.

10 is bounded. The first quadrant is not bounded. The upper half plane is not bounded. The condition for boundedness means that the set is contained in the disc of radius C, as shown on Fig. to. [I, §4] 19 LIMITS AND COMPACT SETS Figure 10 Let f be a function on S, and let a be an adherent point of S. Let w be a complex number. We say that w = limf(z) Z-H ze5 if the following condition is satisfied. Given such that if z E Sand Iz - al < fJ, then f. > 0 there exists fJ > 0 If(z)-wl

The union of the family is the set V consisting of all Z such that Z E Vi for some i E I. We say that the family covers S if S is contained in this union, that is, every Z E S is contained in some Vi' We then say that the family {V;}iel is an open covering of S. If J is a subset of I, we call the family {Vj}jeJ a subfamily, and if it covers S also, we call it a subcovering of S. 9. Let S be a compact set, and let {V;}iel be an open covering of S. Then there exists a finite subcovering, that is, a finite number of open sets ViI"" ,Vi" whose union covers S.