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Unformatted text preview: _.Gedrge M. Bergman Fan 2008, Math 1B Â« .  â€œ 24 0a., 2008
155 Dwinelle Hall â€˜ Second Midterm . 3:10â€”4:00 PM This is a closed hook exam. You are allowed one 2â€”sided 81/2" X 11" sheet of notes. _
Attempt all problems. Write solutions on these sheets. Ask for scratch paper if the
â€™ fronts and backs of these pages are not sufï¬cient; put your name on any such extrasheets 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. . Questionsâ€˜begin on the next sheet. 2 Fill in â€˜your nameand section on this sheet now, but .
"3 'the"page___until the'si nail is. given. At the end ';.o:Â§.the exam, stop Writing and
I \ â€˜ z(Eriialis given,'orfyou Wiillz,â€™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 8:00â€” 9:00 201 El Benjamin Tsou
3102 Etcheverry 9:00â€”10:00 203 D Kiril Datchev
71 Evans 10:0011:00 204 [:1 Benjamin Tsou
3111 Etcheverry 11:0012:00 205 D Harold Williams
75 Evans 12:00â€” 1:00 206 [:1 Koushik Pal
70 Evans 100â€” 2:00 207 1:! Gary Sivek
105 Latimer. 2:00â€” 3:00 208 [:1 Gary SiVek .. [2.3102 Etcheverry 12:00 3:00.? 2141' El Koushik Pa]
85 Evans :15':00â€”"â€˜6:00' 210 [:1 Harold Williams
Other or none [:1 Explain . Leave blank for grading /48 MUTTS Patrick McDonnell ~ â€”â€” m â€˜
mm noel mmwmummm pointsieaeh.)"Ile'ernptIte' the following If an express'iontis undeï¬ned, say so. u work I r . ' answers:
(a) If} f2 d? (a)
(b) â€˜limln__) (2â€+3")_/4"V (b)
(c) 23:0 (2â€f3")/4â€ (c)
(d) The ï¬rst 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 = â€”1 but at no other point. 3, (16 points) Suppose a0, a1, a2, , an, are real numbers,â€™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 anr" convergesabsolutely. â€˜ 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 3:104:00 PM 1. (48 points: 12 points each.) Compute the following. If an expression is undeï¬ned, say so.
(a) [:1 xâ€2 dx Answer: Undeï¬ned (b) limn_>00 (2n+3")/4" Answer: 0 (0) 23:0 (2"+3â€)/4" Answer: 6 (d) The ï¬rst four terms (i.e., the constant term through the x3 term) of the Maclaurin series for (1 +x +x72)â€˜1 Answer: lâ€”x+x3. (You can also show a term â€œ +Ox2 â€, but you donâ€™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 (xâ€”50)'n/50â€. (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 = (â€” l)"
(c) A power series which converges at x = â€”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 22Â°20 an râ€ 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" . Wâ€œ â€˜
(192221,â€œ L Â¥ converges absolutely, r Reminder: The reading for Monday, October 27, is #22 â€˜u â€˜E
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