Review Outline for Math 121 Test 4 11.1 Introduction to Infinite Sequences Be able to write out the first few terms of a sequence, given an nth term formula. Given the terms of a sequence, be able to produce an nth term formula. Sketch the graph of a sequence. Give definition of the following terms and determine if an infinite sequence a) is bounded above/below d) has a limit b) is increasing/decreasing e) is geometric c) is monotone f) is arithmetic Know how to use L’Hôpital’s rule, and the squeeze theorem to determine limits. 11.2 Introduction to Infinite Series Know the difference between a sequence and a series. Know the definition of convergence and divergence of series. Understand and use the Divergence Test (and it’s converse) correctly. Write terms of the partial sum sequence; use sigma notation to represent infinite series Sum geometric sequences, when possible, and know when it is possible. Know what the Harmonic Series is Be able to sum telescoping series. Be able to determine if an infinite series converges or diverges. 11.3, 11.4, 11.5, 11.6 Convergence tests for Infinite series (See summary on next page) 11.7 Power Series Use the ratio test to determine the radius of convergence for a Be able to give the interval of convergence for a series Given a power series for f(x), be able to write the series for f (g(x) ) (ie, if you know the series for sin x, be able to produce the series for sin 2x or sin x2.) 11.8 Taylor and Maclaurin Series Be able to find the Taylor or Maclaurin series of a given function Memorize the Maclaurin series for sin x, cos x, and ex. (Also, geometric series with r = x) Be able to multiply and divide power series; also to calculate limits by using infinite series Summary of Series Known Series ar n1 (aka Geometric Series) converges to S n1 1 n n1 p a , as long as -1 < r < 1 1 r Nth term test 1 n (aka p-series) converges only for p > 1 (aka harmonic series) diverges n1 If lim an 0, then the series diverges If lim an =0, then the series could converge or diverge n n Integral Test Let f(n) = an. If an > 0 f (x)dx converges/diverges, then 1 If If a a a n n1 a If a If diverges n n1 < known convergent series, then n converges n1 n > known convergent series, then no conclusion n < known divergent series, then no conclusion n1 an > known divergent series, then n1 an > 0 converges/diverges n n1 Direct Comparison Test a n1 Limit Comparison Test an > 0 n known n If known n Let lim n an 1 an conv div a n converges n diverges n1 = L, L ≠ 0, L = then 1) bn+1 < bn and 2) lim Ratio Test = L, L = 0, L ≠ then an lim If Alternating Series Test an lim If bn = 0, then (1) n1 a n1 bn converges n1 L If L > 1, then a n diverges n converges n1 If L < 1, then a n1 If L = 1 , then the test fails—must use a different test Growth hierarchy of functions ln n n pos fraction 1 n pos whole number exponentia l n (c ) factorial n! variable base exponential nn