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Unformatted text preview: Practice Test For The Benefit of Mr. Kite David Imberti April 16, 2010 1 Disclaimer The stuff in here should be correct. But I didn’t copy this from somewhere. It’s coming straight from me bashing on a keyboard for the past three or four hours. So if you spot any errors or have any confusion, let me know so I can correct it. Others are studying from this. But other than this worrying disclaimer, I’m pretty sure the stuff in here is spot-on. So don’t worry about it. 2 Warm-up Problems: Not Lava. More like a hot bath in Tokyo. SPOILER WARNING: The following are worked out as example in the discussions that follow. I included more bonus problems after these discussions. Do the following converge or diverge? (0) Σ ∞ n =1 sin ( n ) n 2 (1) Σ e- n n . (2) Σ q n n +1 . (3) Σ n 1 n + n 2 ≤ Σ n 1 n 2 . (4) Σ ∞ n =3 ln ( n ) n . (5) Σ ∞ n =1 (- 1) n 25 e n (2 n +3)! . (6) Σ ∞ n =1 1 ln | n | . (7) Σ ∞ n =1 1 7 n 8 +30000 n . (8) Σ ∞ n =1 (1 + 1 n + 1 n 2 ) 25 e- n . (9) Σ ∞ n =1 (1 + 1 n ) n 2 . (10) Σ ∞ n =1 ln ( n ) n . Are the following Absolutely Convergent, Conditionally Convergent, or Divergent? (11) Σ ∞ n =0 10- n . (12) Σ ∞ n =3 (- 1) n ln ( n ) n . (13) Σ ∞ n =1 (- 1) n n +1 nln ( n ) . (14) Find the minimal number of terms one needs to sum to approximate the convergent alternating series Σ ∞ n =2 (- 1) n n ! to an error of less than or equal to 1 120 . Find the Interval of Convergence of the following. (15) Σ ∞ n =0 x 2 n (2 n +2)! . (16) Σ ∞ n =0 n ! x n . (17) Σ ∞ n =0 | x- 3 | n . (18) Simplify the sum Σ ∞ n =0 x n + Σ ∞ n =1 x n- 1 . 1 Find the Power Series Expansions of the following. (19) 3 3- x . (20) ln (1- x ). 3 Introduction In the previous practice exam I had a small section including loads of tips and hints. This time I noted that the book has a pertty good summary on this. Go to page 721. Read that page. That page pretty much contains all the advice I would give you. The only other advice is as follows ... 4 WHAT DO I DO? Q: WHAT’S THIS EXAM ABOUT? WHAT DO I DO, THERE’S LIKE A MILLION TESTS, AND I AM SO OVERWHELMED. A: It’s about series. The exam is on Tuesday. Calm down, you got a week. Also, there are only 7 major tests, but I’m going to make it basically 5. (you’ve also got stuff about alternating series and power series and radius of convergence; but that’s only 3 things and I’ll cover it in the next section) Q: COULD YOU GIVE ME THE RUN-DOWN? THE BOOK IS LIKE A THOUSAND PAGES LONG. A: Here are the tests, in the order I’d try them out: (0) Do the terms go to 0? (1) Comparison Test (a) With a Geometric Series (b) With a p-series (2) Ratio Test (3) Limit Comparison (4) Root Test (5) Integral Test Q: OH MAN, THIS IS SO CONFUSING, I THOUGHT THE RATIO TEST AND LIMIT COMPARISON TESTS WERE THE SAME? WHAT’S THE TEST YOU NEED THE LIMIT > 0 AND WHICH ONE DOES ALL THAT STUFF WITH < 1 ,> 1 , = 1?? ARGH....
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This document was uploaded on 01/25/2012.
- Spring '09