By A.M. Fink

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Let {aj } be a real sequence. Prove that if lim inf aj = lim sup aj then the sequence {aj } converges. Prove the converse as well. 5. Let a < b be real numbers. Give an example of a real sequence whose lim sup is b and whose lim inf is a. 6. How is lim sup(aj + bj ) related to lim sup aj and lim sup bj ? 7. How is lim sup(aj − bj ) related to lim sup aj and lim sup bj ? 8. What is lim supj→∞ cos j? What is lim supj→∞ sin j? 4. SOME SPECIAL SEQUENCES 35 9. What is lim supj→∞ ln j? 10. Prove that, if lim inf aj < lim sup aj , then the sequence {aj } cannot converge.

3. 18. 4. Let a1 , a2 > 0 and for j ≥ 3 define aj = aj−1 + aj−2 . Show that this sequence cannot converge to a finite limit. 5. Suppose a sequence {aj } has the property that, for every natural number N , there is a jN such that ajN = ajN+1 = · · · = ajN+N . In other words, the sequence has arbitrarily long repetitive strings. Does it follow that the sequence converges? 6. Let γ be an irrational real number and let aj be a sequence of rational numbers converging to γ. Suppose that each aj is a fraction expressed in lowest terms: aj = αj /βj .

This gives another nice way to think about this example. 24 Let us apply the Cauchy Condensation Test to the series ∞ j=1 1 . j(log2 j)2 This leads us to examine the series ∞ k=1 2k · 1 = 2k · k 2 ∞ k=1 1 . k2 By the preceding corollary, this is a convergent series. So the original series converges. 25 (The Root Test for Convergence) Consider the series ∞ aj . j=1 If 1/j lim sup |aj | j→∞ then the series converges. 2. ELEMENTARY CONVERGENCE TESTS 53 Proof: Refer again to the discussion of the concept of limit superior in Chapter 2.