By Andras I. Stipsicz Robert E. Gompf

The previous 20 years have introduced explosive progress in 4-manifold idea. Many books are at the moment showing that method the subject from viewpoints reminiscent of gauge conception or algebraic geometry. This quantity, although, deals an exposition from a topological viewpoint. It bridges the distance to different disciplines and offers classical yet very important topological thoughts that experience now not formerly seemed within the literature. half I of the textual content offers the fundamentals of the idea on the second-year graduate point and gives an outline of present study. half II is dedicated to an exposition of Kirby calculus, or handlebody idea on 4-manifolds. it really is either easy and complete. half III bargains extensive a extensive variety of themes from present 4-manifold examine. subject matters comprise branched coverings and the geography of complicated surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. purposes are featured, and there are over three hundred illustrations and diverse workouts with ideas within the publication.

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Deciding whether the corresponding series converge or diverge is a bit harder because no convenient formulas for Sn are available. Co nvergence a n d di v e r g e n c e : g r aph i c al v i e w s For a series ak , plotting both the terms {ak } and the partial sums {Sn } on the same axes illustrates the connection between the two— and sometimes suggests whether the series converges or diverges. The two series 1/k and 1/k 2 generate Figures 1(a) and 1(b), respectively. 5 1 {ak} {ak} 10 20 30 40 (a) Terms and partial sums for 10 50 1/k 20 30 (b) Terms and partial sums for 40 50 1/k 2 FIGURE 1 Terms and partial sums for two series The two pictures give different impressions.

N Sn = a1 + a2 + a3 + · · · + an−1 + an = ak . (3) k=1 The nth tail, denoted by Rn , is the (inﬁnite) sum of all terms beyond index n: ∞ Rn = an+1 + an+2 + an+3 + · · · = ak . k=n+1 As the notation Rn suggests, the nth tail is a remainder—what’s left after adding terms through index n. In symbols, ∞ n ak = k=1 ak + k=1 ∞ ak = Sn + Rn . ) The crucial deﬁnition of convergence involves the partial sums Sn : D E F I N I T I O N If lim Sn = S, for some ﬁnite number S, then the series n→∞ ∞ k=1 ak converges to the limit S.

Convergent but not monotone. 20. bounded but not monotone. C H A P T E R 11 554 Inﬁnite Series 21. monotone but not convergent. 22. nonincreasing and unbounded. 25. 7777777, . . has a limit. 23. nonincreasing and convergent. 26. Suppose that lim ak = L, where L is a ﬁnite number, and that k→∞ the terms of the sequence {bk } are deﬁned by bk = L − ak . Explain why lim bk = 0. 24. unbounded but not monotone. k→∞ FU EX RT H ER ER CISES In Exercises 27–32, ﬁnd the limit of the sequence or explain why the limit does not exist.