1. If∑anis a divergent series with positive terms, then(a)limn→∞an6= 0.(b)∑(-1)nandiverges, too.(c) Ifbn< anfor allnthen∑bnconverges.(d) None of the above.

∑anconverges, which conditionon L would guarantee that∑bnconverges?

3. If∑(-1)nanis an alternating series and limn→∞an= 0, which further condition wouldguarantee that the series converges by AST?

4.TRUE or FALSE: If∑anand∑bnare both convergent series with positive terms,[1]then∑anbnis also convergent.5.TRUE or FALSE: If∞Xn=1anconverges then∞Xn=1(1 + cos(an))nconverges.[1]

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(SA) Short answer questions, marks only awarded for a correct final answer, you do not need toshow any work.(a)How many terms would you need to use to estimate the sum of∞Xn=1(-1)nnwith an[2]error of at most12019?(b)If you used the first 333 terms to estimate the sum of∞Xn=1(-1)n2nnen, would it be an[2]overestimate or an underestimate?(LA) The remaining questions are long answer questions, please show all of your work.1.Determine all values ofp∈Rfor which∞Xn=21n(ln(n))pconverges. You can use without[3]proof thatf(x) =1x(ln(x))pis continuous, positive, and decreasing eventually for allp∈R.

2. Determine whether the following series are convergent or divergent.(a)∞Xn=1√n2+ 3n+ 1n5-n4+n[3](b)∞Xn=2(-1)nln(n)[3](c)∞Xn=1lnn(n+ 2)(2n+ 3)2[3]

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