By Singer I.

ISBN-10: 0471160156

ISBN-13: 9780471160151

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**Extra info for Abstract convex analysis**

**Sample text**

The series is divergent for |x| > �. For |x| = � a special investigation is needed, in which one may possibly apply Abel’s theorem (see below). In general, convergence at the end points is only of limited interest, even though this problem can often be met with at examinations. The radius of convergence � is formally deﬁned by �∞ � � �� � � n an x is convergent . |x| � �= � n=0 However, one rarely applies this deﬁnition. Instead one typically derive the radius of convergence from the root test or the ratio test: 1) Preparations (in order to avoid errors).

2) If ‘yes’, ﬁnd the pointwise limit function f (x), and proceed with c) below. c) If the pointwise limit function f (x) is not continuous, it follows from a) and a theorem in the next chapter that (fn (x)) is not uniformly convergent. If instead f (x) is continuous proceed with d) below. d) Remove the variable x by an estimate like |fn (x) − f (x)| ≤ sup |fn (x) − f (x)| ≤ an , n ∈ N, x∈I where an is not depnding of x. If an → 0 for n → ∞, then (fn ) is uniformly convergent with the limit function f (x).

Com 53 Calculus 3b General series; tricks and methods in solutions of problems The ﬁrst majoring series is divergent (hence no conclusion, because the estimate is too crude), and the second one is convergent, hence the trigonometric series is uniformly convergent. (The ﬁrst majoring series is actually taken from an examination, where the conclusion unfortunately became wrong). 2 Let every function fn (x), x ∈ I, be continuous. ) If f (x) = uniformly convergent, then the sum function f (x) is continuous.

### Abstract convex analysis by Singer I.

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