MATH 314 001 – Test #2 – 11/10/06 1Sat Nov 11 07:24:36 MST 2006 /m314.fa06/handouts314/t2 314 B10/t2 314 B10This test has pages 1 – 2. Take a moment to make sure you have them all.1 Does∞Xn=2(−1)nn(−5)nconverge? Justify your answer.2 Definition question about Cauchy Sequences:(a) State the basic definition of Cauchy Sequence.(b) Give an example of a sequence which is not a Cauchy sequence. Explain briefly whyyour example is not a Cauchy sequence.(c) State the Cauchy Criterion for infinite series.3 Definition question about lim sup:(a) State the basic from-scratch definition of lim sup, the MATH-314 version.(b) State one of the theorems that provides an alternative characterization. That is, statea theorem which could be used as a definition of lim sup because it provides a logicalequivalence to the from-scratch definition.(c) Give an example of sequences (sn) and (tn) such thatlim sup(sn+ tn) < lim sup(sn) + lim sup(tn).Explain briefly why your example does this.PTO: more on other side.MATH 314 001 – Test #2 – 11/10/06 24 Proof-writing problems – choose just ONE to write up during this test period:(a) Write a nice proof of the theorem: if S ⊆ R and sup(S) is in R but not in S, then amonotone-increasing sequence (sn) can be found in S such that lim(sn) = sup(S).(b) State and prove the Monotone Sequence Theorem. To do this problem, it is enough towrite down just the Bounded Monotone-Increasing Sequence Convergence Theorem,and show how it depends on more basic principles.(c) State and prove the theorem about what happens if lim sup(an) = lim inf (an),where lim sup(an) and lim inf(an) are both in R.(d) Write down how the Root Test indicates convergence. Then write a proof of theresult.5 Take-home problems – do both for Monday.(a) Suppose that∞Xk=1akis a series for which∞Xk=1|ak| satisfies the Cauchy Criterion forinfinite series. Show that∞Xk=1akmust also satisfy the Cauchy Criterion for infiniteseries.(b) We say that (an) ∼ (bn) for sequences (an) and (bn) if∀ε > 0 ∃N ∀nn > N =⇒ |an− bn| < ε.Prove that, if (an) ∼ (bn) and (an) converges with limit A, then also lim(bn) = A.Sat Nov 11 07:24:36 MST 2006 /m314.fa06/handouts314/t2314 B10/t2 314