E. A rational function. is a function of the form e] = p(}fq(s} dened for :s E IR where pie} = Eli-La Era-Ii, sis) = ZED fir-'3' are both polynomials Suppose an = pinliei} where n.) 3% IE] and u) 5-5 I] for ever},r n. Prove that n+1 -n . pin + 1} _ sin} nLc-o Using ['Jorollai'g.r 12.3, conclude that the root and ratio tests always fail to provide information when on is a rational function. Notes: For the sake of discussion, take p{:s),q(m} to he polynomials of degree A: and m respectiver in your solution so that 111(3) 2 Ella] crj-erj, slim} = EEC, lm: with or;a a": D, lm 3% i]. This result shows that there is a broad class of series for which the root and ratio tests tail to prewide information, e.g. Z Eli5:1, E \"351312, etc. So in these cases, you must use the comparison test or another test to deduce convergence or divergence of the series