Math 1B Practice Final Rob Bayer August 11 2008 The actual final will cover the entire course and be 50 integration and ODEs and 50 sequences and series See previous practice and actual midterms for problems covering material from the first 2 3 of the course n P x 1 n 1 Find the radius and interval of convergence for n 1 2 3 n 1 2 a Find the Taylor series for f x sin x centered at cos 4 1 2 4 In case you ve forgotten sin 4 b Find the ROC and IOC for your taylor series from a c Prove that your taylor series converges to sin x for all x 3 Find the solution to the initial value problem y 00 xy 0 y 0 y 0 1 y 0 0 0 Your answer may involve power series if you wish 4 True False Justify your answers with theorems tests or counterexamples P P a If cn xn has radius of convergence R then cn Rn converges conditionally P P b If cn 3n converges then 1 n cn 3n converges P P c If cn 3n converges then 1 n cn 2n converges P P P d If an and bn are conditionally convergent then an bn is too P P 2 e If an is absolutely convergent then an is too 5 Determine whether each of the following series are absolutely convergent conditionally convergent or divergent a X tan 1 n n3 X n b e n2 X n c 2 5 8 3n 2 X 1 n d n ln n X 1 e 1 en X 1 1 f n5 7 n 1 5 7 X cos n g n ln n 2 X 3n h 1 n 3 n X 2 3n i nn X n 2 j ln n Adapted from past midterms from various professors n 1 k n X n n l 1 2 X n 6 Using whatever means you want find the Maclaurin series for each of the following functions a b ln 1 x x x 1 x3 2 ex c R d x2 3 x
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