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Unformatted text preview: Math 113 Exam 3 Practice Exam 3 will cover 11.211.11. Note that even though 11.1 was tested in exam 2, you will still need to know about sequences. This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second gives a number of practice questions for you to work on. The third section give the answers of the questions in section 2. Review Sequences Sequences were tested in Exam 2, but you will still need to deal with them. For example, in the limit comparison test, or the Ratio or Root test, you will need to be able to find limits of sequences. So I am repeating some things here that you were given for the last exam. Some theorems may be of help here: 1. If a sequence converges, it is bounded. 2. If a sequence is bounded and is (eventually) increasing or decreasing, then it converges. 3. If a sequence { a n } matches a function f (i.e. f ( n ) = a n ) and lim n →∞ f ( x ) = L, then the limit of the sequence is also L . Rule 3 is useful because we can use everything we know about limits of functions to find limits of sequences. Since L’Hopital’s rule is one of them, you should expect to use it. There are other rules about sums of sequences and products of sequences, etc. You are advised to review them in the text. Important Sequences Some limits occur often enough that it is advisable to know about them in advance. For example, it is important to know the following: 1. If c is a real, positive number, then c 1 /n → 1. 2. If c is a real, positive number, then 1 n c → 0. 3. c n n ! → 0. 4. n 1 /n → 1. 5. ( 1 + c n ) n → e c . All of the above limits except the third can be proven using L’Hopital’s rule. The third is a bit tricky but can be done by noticing that the ( n + 1)st term of the series is c n + 1 times the n th term. If you encounter these limits in a problem, you are welcome to use what you know about them and move on. Series In this section we learned about convergent and divergent series. A series converges if the sequence of partial sums converge. There are some particular types of series that we learned about: Geometric Series ∞ ∑ n =0 ar n . We learned that the geometric series converges to a 1 − r if  r  < 1 and diverges other wise. We saw several applications where we could write a problem in terms of a geometric series. Harmonic Series ∞ ∑ n =1 1 n . We saw by examination of the partial sums s 2 n that this diverges. The integral test also shows the divergence of this series. 1 Alternating Harmonic Series ∞ ∑ n =1 ( − 1) n +1 n We know by a later test that the alternating Harmonic series con verges. We know by the Maclaurin series of ln(1 + x ) that it converges to ln(2)....
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 Fall '00
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 Math, Calculus, Geometric Series, Formulas, Mathematical Series

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