# Math 172 : Midterm 3

Date: Tuesday, April 14, 6:00-7:50pm

 Summary Points: This exam requires a fair amount of memorization but will not require  hard calculations. The sequences an defining the series are all simple (or even very simple) so none of the limit calculations are hard. For the Integral test, you may have to evaluate an improper integral by substitution. You may need to take multiple derivatives of simple functions for Taylor series. By far, however: Know your convergence theorems, how to use them, how to verify hypotheses if needed and how to state your answer clearly. 1) Exam Topics  Geometric and Telescoping Series                                              (section 10.2) Divergence Test                                                                       (section 10.2) Integral Test and p-series                                                         (section 10.3) Comparison and Limit Comparison Tests                                    (section 10.3) Alternating Series Test and Error Bounds                                   (section 10.4) Absolute and Conditional Convergence                                      (section 10.4) Ratio and Root Tests                                                                (section 10.5) Power Series and radius/interval of convergence                        (section 10.6) Taylor series expansion (Theorem 1, pg 592).                            (section 10.7) Shortcuts for Taylor series (pg 594-597) and limits                     (section 10.7) 2) There is no question specifically on the sequence material in section 10.1. However, you will be required to take limits of sequences {an} when using tests like the Ratio, Root and Limit Comparison tests. 3) Unless otherwise stated you must verify all test hypotheses for full credit as in Comparison tests, Integral test, alternating series test etc. For some questions you will be required (at some point) to state and verify hypotheses. In all instances clearly state your conclusions. Just writing "converges" is not sufficient But "the series converges by the Limit Comparsion Test" is sufficient since it says what is converging and by what test. 4) There will be a 15 point Theorem and Concepts problem with 5 short True/False questions like these (or at the end of this). 5) Miscellaneous remarks: A formula sheet will be provided for Maclaurin Series -  formula sheet below. You are still required to know Theorem 1 in (10.7) for finding the Taylor series centered at x=a :   If you use L'Hospital's rule one must take the limit of the associated real function, i.e. f(x)=sin(x)/x versus sin(n)/n. Failure could cost a point. Same applies when using the integral test, i.e., you can integrate the function f(x)=ln(x)/x but not the sequence an=ln(n)/n. Many exam questions are minor variants of homework problems. Generally speaking the HW is a good guide to the kinds of problems on the test. You should be able to do the exam in 60-75min but will have 1hr and 50min 6) Remember our tutoring resources - both the Math Learning Center and review sessions held by Corinne Casolara (SSC).

Formula Sheet that will be attached to Midterm:

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