MATH 355 Final: MATH 355 Amherst F13M355 2802 29ManackFinal
15 Feb 2019
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Amherst college, math 355 final, fall "13, manack. Xk=0 and assume that f, g converge pointwise on v (0) ( > 0). If f = g on v (0), prove ak = bk for all k n {0}. akxk, g(x) = Xk=0 (b) recall, the fibonacci sequence is de ned recursively by a0 = 0, a1 = 1, and an+1 = an + an 1 for all n n: prove that (an) is a positive sequence. Then prove quickly that (an) is in- creasing: now prove quickly that an+1/an 2 for all n n, use the ratio test to prove that the power series f (x) = 1 x x2 for all |x| < 1/2. decomposition: factor 1 x x2 as (1 ax)(1 bx), then perform the partial fraction x. Nding a, a, b, b r. determine the power series of the right hand side, as well as its radius of convergence: use part (a) to deduce binet"s formula:
