Dr. Z’s Math152 Handout #11.1 [Sequences]By Doron ZeilbergerProblem Type 11.1a: Determine whether the sequencean= f(n)(where f(x) is a ‘nice’ function) converges or diverges. If it converges find its limit.Example Problem 11.1a: Determine whether the sequencean=3 + 5n2n + n2,converges or diverges. If it converges find its limit.Steps Example1. Replace the n by x and use your CalcI expertise (you can consult:http://www.math.rutgers.edu/˜zeilberg/sod/ho2.pdf ) to computelimx→∞f(x) .1. To computelimx→∞3 + 5x2x + x2.note that when x gets bigger and bigger,3 is insignificant next to 5x2and x is in-significant next to x2(what I call ‘forgetabout the little ones’), so the limit can bereplaced by the simpler limitlimx→∞5x2x2= 5 .2. If that limit diverges, then the se-quence diverges. If it converges then thesequence converges and the value of thelimit is the same.2. The above limit exists and is equal to5, soAns.: The sequence converges and its limitequals 5.Problem Type 11.1b: Determine whether the sequencean= (−1)nf(n)(where f(x) is a ‘nice’ function) converges or diverges. If it converges find its limit.1Example Problem 11.1b: Determine whether the sequencean=(−1)n(3 + 5n2)n + n2,converges or diverges. If it converges find its limit.Steps Example1. Replace the n by x and use your CalcI expertise (you can consult:http://www.math.rutgers.edu/ zeilberg/sod/ho2.pdf ) to computelimx→∞f(x) .1. To computelimx→∞3 + 5x2x + x2,note that when x gets bigger and bigger,3 is insignificant next to 5x2and x is in-significant next to x2(what I call ‘forgetabout the little ones’), so the limit can bereplaced by the simpler limitlimx→∞5x2x2= 5 .2. If that limit diverges, then the se-quence definitely diverges. But even itconverges to something other than 0, thesequence, because of the (−1)nin it, stilldiverges. Only if the above limit is 0 doesthe sequence converge, and then the limitof the sequence is 0.2. While the above limit exists, it is notequal to 0, soAns.: The sequence diverges, because inthe long-run it is like 5, −5, 5, −5, . . . so itcan’t make up its mind.Problem Type 11.1b’: Determine whether the sequencean= (−1)nf(n)(where f(x) is a ‘nice’ function) converges or diverges. If it converges find its limit.Example Problem 11.1b’: Determine whether the sequencean=(−1)n(3 + 5n2)n + n3,converges or diverges. If it converges find its limit.Steps Example21. Replace the n by x and use your CalcI expertise (you can consult:http://www.math.rutgers.edu/ zeilberg/sod/ho2.pdf ) to computelimx→∞f(x) .1. To computelimx→∞3 + 5x2x + x3,note that when x gets bigger and bigger,3 is insignificant next to 5x2and x is in-significant next to x3(what I call ‘forgetabout the little ones’), so the limit can bereplaced by the simpler limitlimx→∞5x2x3= limx→∞5x= 0 .2. If that limit diverges, then the se-quence definitely diverges. But even itconverges to something other than 0, thesequence, because of the (−1)nin it, stilldiverges. Only if the above limit is 0 doesthe sequence converge, and then the limitof the sequence is 0.2. Not only does the above limit exist, itis equal to 0! soAns.: The sequence converges to
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