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EXERCISE 1. (5+3=8 marks) Let (an) be a sequence of resal numbers. 1. Prove that if an converges absolutely, then converges. Solution: 2. Show that this result does not hold in general if we only assume that In converges conditionally. Solution: EXERCISE 2. (2+4+6+5=17 marks) This exercise revolves around Abel's theorem for series. Theorem 1 (Abel) Let (an) and (on) be sequences of real numbers such that (an) 18 non-increuzing and lends to 0, there erists M ER such that, for any 72 € N. Li=1 bel < M. Then the series and converges. 1. Show that the alternate series test can be deduced from Abel's theorem. 2. Using Abel's theorem, show that converges (hint: we have cos(n) = where R denoles the real part of a complez number, and Solution: Let (an) and (bz) be sequences of real numbers. The Abel's formula resads, for 2. =(a1-a2)6, + (a2->a3)(bi +b2) + (as an)(b1 + 62 bs) n=1 +... + (ap-1 -an)(b1 +...... (1) 3. Prove Abel's formula. You can for example notice that the right-hand side of (1) is p-1 (ak-ak+1) (") - 66 k=1 I=1 and prove by induction on P that this is equal to Solution: 4. Use Cauchy's criterion for series and Abel's formula to prove Abel's theorem. EXERCISE 3. (MTH3140 only) (3+4=7 marks) A sequence (an) is pseudo-Cauchy if for any E> 0 there exists N. € N such that an+1 -an s E for all 72 N. 1. Show that a sequence that converges is pseudo-Cauchy. 2. Is a pseudo-Cauchy sequence always converging? If yes, provide a proof. Otherwise, provide a counter-example. Solution:

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