EXERCISE 1. (5+3=8 marks) Let (an) be a sequence of resal numbers.
1. Prove that if
an converges absolutely, then
2. Show that this result does not hold in general if we only assume that In converges conditionally.
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
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)
+... + (ap-1 -an)(b1 +...... (1)
3. Prove Abel's formula. You can for example notice that the right-hand side of (1) is
(ak-ak+1) (") - 66
and prove by induction on P that this is equal to
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.
Is a pseudo-Cauchy sequence always converging? If yes, provide a proof. Otherwise, provide a counter-example.
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