This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 spoton. So don’t worry about it. 2 Warmup 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 RUNDOWN? 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 pseries (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....
View
Full
Document
This document was uploaded on 01/25/2012.
 Spring '09

Click to edit the document details