Calculus II PLTLFall 2014Worksheet 8These problems are to be do ne without the use of a calculator unless otherwisespecified.Before beginn i ng to work on the following problems , explain what these state-ments mean:1) The sequence {sn}∞n=0converges.2) The sequence {sn}∞n=1is divergent.3) The seriesP∞n=2anis convergent.4) The seriesP∞n=3andiverges.1) (Scribe) Find a formula for the nth partial sum of the seriesP∞n=12n2+4n+3and useit to fi nd the series’ sum, if it converges.2) (Round Robin) Match each of the general terms from a sequence ((a)-(e)) with thecorrect description of its behavior as n → ∞ ((I)-(V)).(a) an= (−1/5)n(I ) Diverges (oscil lates)(b) bn= 1/[ln(n) − ln(n + 1)] (II ) Diverges (to − ∞)(c) cn= 2 + (−1)n(III ) Converges to 0(d) dn= (1 −1n)n(IV ) Converges to 1(e) en= (n + [−1]n)/n (V ) Converges to 1/e3) (Pairs) Let an= (n − 10 0)2/1000.(a) Compute the value of each term indicated below.a1=a2=a3=a4=a96=a97=a98=a99=Are the terms increasing or decreasing? Do the terms seem to be tending to a limit?(b) Now compute the values o f these terms.a101=a102=a103=a104=What is going on here? Does this sequence seem to be converging now?(c) Compute limn→∞an. Does this tell you whether {an} converges or diverges?(d) Let f(x) = (x − 100)2/1000. Then an= f(n) for all positive integers n. Sketchthe graph of f. How does this help to explain the answers you got in (a)-(c)?(e) This example s hows that it can be useful to look at a real-valu ed fu nction tounderstand the beh avior of a sequence. One way that the behavior of functions that matchup with terms o f series at integer values can sometimes be determined is with the IntegralTest, which can ’t be applied to this series. Why not?4) (Scribe) Determine whether the seriesP∞n=31/[n ln n(ln(ln n))2] converges or di verges.5) ( ) Use a geometri c series to rewrite the repeating decimal3.142857142 857142857 . . . as a rational number.6) (Pairs) Suppose thatPanconverges with an> 0 for all n. Decide i f each of thefollowing series converges, diverges, or if ther e is not enough information provided.(a)Pan/n(b)P1/an(c)Pnan(d)P(an+
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