F08_Second_Midterm-G.Bergman

F08_Second_Midterm-G.Bergman - _.Gedrge M. Bergman Fan...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 4
Background image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: _.Gedrge M. Bergman Fan 2008, Math 1B « . - “ 24 0a., 2008 155 Dwinelle Hall ‘ Second Midterm . 3:10—4:00 PM This is a closed hook exam. You are allowed one 2—sided 81/2" X 11" sheet of notes. _ Attempt all problems. Write solutions on these sheets. Ask for scratch paper if the ’ fronts and backs of these pages are not sufficient; put your name on any such extra-sheets and hand them in with yourexam. I > Credit for an answer may be reduced if a large amount of irrelevant or incoherent material is included along with the correct answer. . Questions‘begin on the next sheet. 2 Fill in ‘your nameand section on this sheet now, but . "3 'the"page___until the'si nail is. given. At the end ';.o:§.-the exam, stop Writing and I \ ‘ z(Eriialis given,'orfyou Wiillz,’lOSe'points. V Your name . _ Sections: Mark yours with X. .(Note that they are listed in order of hour, not sectionfnumber.) usual place, hour,(MW), Sec. TA 171 Stanley 8:00— 9:00 201 El Benjamin Tsou 3102 Etcheverry 9:00—10:00 203 D Kiril Datchev 71 Evans 10:00-11:00 204 [:1 Benjamin Tsou 3111 Etcheverry 11:00-12:00 205 D Harold Williams 75 Evans 12:00— 1:00 206 [:1 Koushik Pal 70 Evans 100— 2:00 207 1:! Gary Sivek 105 Latimer. 2:00— 3:00 208 [:1 Gary SiVek .. [2.3102 Etcheverry 12:00 3:00.? 2141' El Koushik Pa] 85 Evans :15':00—"‘6:00' 210 [:1 Harold Williams Other or none [:1 Explain . Leave blank for grading /48 MUTTS Patrick McDonnell ~ —— m ‘ mm noel mmwmummm pointsieaeh.)"Ile'ernptIte' the following -If an express'iontis undefined, say so. u work I r . ' answers: (a) If} f2 d? (a) (b) ‘limln__) (2”+3")_/4"V (b) (c) 23:0 (2”f3")/4” (c) (d) The first four terms (i.e., the constant term through the (d) . x3 term) of the Maclaurin series for (1 +x +x2)_1 2. (36 points, 12 points each) For each of the items listed below, either give an example with the property stated; or give a brief reason why no such example exists. (If you give an example, you are not asked to show that it has the asserted property.) (a) A power series whose interval of convergence is the open interval (0, 100). (b'), A bounded sequence of real nurnbers a9, a1, a2, , which does not converge. (c) A power series which converges at x = —1 but at no other point. 3, (16 points) Suppose a0, a1, a2, , an, are real numbers,’and that r and s are nonzero real numbers with Irl < sl. Prove that if the sequence. Idol, Ia] sl, laz $2], , Ian 3"], is bounded, then the series 23:0 an-r" converges-absolutely. ‘ I This was part of a lemma that I proved in class, in proving the theorem on radii of convergence of power series; so of course you cannot quote that lemma, or the theorem on radii of convergence, in proving this. But you can use any of the earlier results we had concerning sums of series. Though remembering the lecture might help yOu do this problem, it is not necessary. The proof is a straightforward application of earlier results in the text. George M. Bergman Fall 2008, Math 1B 24 Oct., 2008 155 Dwinelle Hall Solutions to the Second Midterm 3:10-4:00 PM 1. (48 points: 12 points each.) Compute the following. If an expression is undefined, say so. (a) [:1 x”2 dx Answer: Undefined (b) limn_>00 (2n+3")/4" Answer: 0 (0) 23:0 (2"+3”)/4" Answer: 6 (d) The first four terms (i.e., the constant term through the x3 term) of the Maclaurin series for (1 +x +x72)‘1 Answer: l—x+x3. (You can also show a term “ +Ox2 ”, but you don’t need to.) 2. (36 points, 12 points each) For each of the items listed below, either give an example with the property stated, or give a brief reason why] no such example exists. (If you give an example, you are not asked to show that it has the asserted property.) (a) A power series whose interval of convergence is the open interval (0, 100). Answer: 2220 (x—50)'n/50”. (For this and the other parts of this question, there are other examples you could have given than the ones shown here.) (b) A bounded sequence of real numbers a0, a1, a2, ..., which does not converge. Answer: an = (— l)" (c) A power series which converges at x = —1 but at no other point. Answer: 27:0 n! (x+1)". 3. (16 points) Suppose a0, a1, a2, ..., an, are real numbers, and that r and s are nonzero real numbers with lrl < Prove that if the sequence laol, lal sl, |a2 s2], ..., Ian snl, is bounded, then the series 22°20 an r” converges absolutely. This was part of a lemma that I proved in class, in proving the theorem on radii of convergence of power series; so of course you cannot quote that lemma, or the theorem on radii of convergence, in provingthis. But you canuse any of the earlier results we had concerning sums of series. v- :1 Though remembering the lecture might help you do this problem,it is not necessary. The proof is a straightforward application of earlier results in the text. ' Answer: The boundedness of the sequence of terms an s" I means that there exists a constant C 2 0 such that Ian snl < C for all n. Now [an rnl = [an s"! lr/sl" S C Ir/sl", so the series 22:0 [an rnl has each term S the corresponding term of the series 2220 C Ir/sln. The latter is a geometric series with ratio Ir/sl < 1, and S0 converges. So 2:120 Ian rnl converges by the comparison test, which says that 22:0 an r" . W“ ‘ (192221,“ L ¥ converges absolutely, r Reminder: The reading for Monday, October 27, is #22 ‘u ‘E i , 1 l l i ...
View Full Document

This note was uploaded on 02/19/2012 for the course MATH 1 taught by Professor Wilkening during the Spring '08 term at University of California, Berkeley.

Page1 / 5

F08_Second_Midterm-G.Bergman - _.Gedrge M. Bergman Fan...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online