# Marks determine whether the series 1 x n 1 2 n e n 2

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11.[7marks]Determine whether the series1Xn=12nen2is convergent or divergent. Name any test that you useand justify your answer fully.12.[9marks]Determine whether the series1Xn=3(-1)nlnnis divergent, conditionally convergent, or absolutelyconvergent. Name any test that you use and justify your answer fully.
MA104 - Final ExaminationPage 5 of 1013.[13marks]Find the radius of convergence and the interval of convergence of the seriesName any test that you use and justify your answer fully.
MA104 - Final ExaminationPage 6 of 1014.[8marks]Find a power series representation for the functionf(x) =x2ln(4 +x) and state the radius ofconvergence.
MA104 - Final ExaminationPage 7 of 1015.[12marks]Letf(x) = 1/x.(a) Find the Taylor series offcentered atx= 1.(Assume thatfhas a power series expansion. Do not show thatRn!0.)(b) Use the Taylor series offto approximateZ1.11dxxcorrect to within an accuracy of 0.0001.Justify your answer fully.Over
MA104 - Final ExaminationPage 8 of 1016.[8marks]Letf(x, y) =x2+xy+y3.(a) Find the equation of the tangent plane to the graph offat the point (1,2).(b) Use the linearization offor dierentials to approximatef(1.1,1.9).
MA104 - Final ExaminationPage 9 of 10