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# Exam1Solutions - Exam 1 Math 408D Name Spring 2009 TA...

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Unformatted text preview: Exam 1 Math 408D Name Spring 2009 TA Discussion Time: TTH You must Show sufﬁcient work in order to receive full credit for a problem. Do your work on the paper provided. Write your name on this sheet and turn it in with your work. Please write legibly and label the problems clearly. Circle your answers when appropriate. No calculators allowed. 1. Evaluate each limit or explain clearly why it does not exist. 1 (a) (14 points) lim n2 sin <—) n—wo 7L (b) (14 points) lim (1:2 > m—>0+ I," 1 2. (14 points) Evaluate /0 Walt or show that it diverges. 0" H2 k+3 3. (14 points) Find the sum of the series Z ( ) k:1 3k or show that it diverges. 4. Determine whether each of the series below converges absolutely, con— verges conditionally, or diverges. State what tests you use and verify that the hypotheses are satisﬁed. °° 2 ‘ k: (a) (14 points) E \$— 1621 (b) (14 points) fix—W“ <\$>k k:0 5. (16 points) Find the radius and interval of convergence of the power series below. Justify all your conclusions. 0° (-3)’“(\$- 2V“ ; k+\/E Bonus (5 points): Assume that Z ak and Z bk are positive termed series and that both converge. Explain clearly why 2 akbk must converge. M4080 SYw/w’ W'W \m Jim; “29““ 3/2”»; MCV“) V\—7°" IA"; 09 M Cody“). ("l/"t3 \A'Vd“ (L3 M10453“ y-ao ‘/ M 31?.“ 7‘ _ y~70+ ”WC L > ’ éﬁvg} L'H - (i) 1. Um} f: 1. (+2." (3 2 Z) Y’QD I 2 2" \$321“ (ﬁ>%z:6+za>sz—%L:%ﬁ I Y \/ r SO Mir £3 >¥J gym 1 ‘xﬂo y“ L 7; 81/2, ’1’“ : SL *2”; Jy 0 (way My"; 06*" “/3 \ 4L1" _ * .L I w 4*" ' if“ 83 7; -, WA 5 SHOW :MW J. <_ l 5‘ ‘15 (I Vat/L— Y ”I '_ " 4" :2 '96 La»; 8 4L2" y c I M K may“ W’ ﬁz+3 3 i :33?"— lg 41 ﬁll/WC" Wﬂc wa MED 3 (an {n+4 k _z_ ahH _' Six“ : LIA-+3 = 3‘ ”J J\Y‘S+'JW r: 71’“— ' 3 (4.) , K m : (:2 : “0/3 . So WW»: WW (8‘ “— 3 a ”/3 ”/5 :9 lrtoWJﬂ? “"51”” 1/ = /r I“ 1w \./ /\"\ 3x x :N s/ V 1-" S + +§ ‘/ \./ H ma WSW“ (n+I)+W—' " : \$h<’2" Jib aux WW BW'Z‘ 41‘ 30 {D W WW \X'LX‘ /\$ ‘/ @Ao/AM‘O 4c» 5’ ‘8 3 up W Qwﬂgomb JWLL WW} 0/?) wk WW Y:.'2>(‘/3 9.41 X’D’V3 \ 06 ‘L 0"" W f0.m ‘6‘ (r33tCQ-‘V3’L>L £1),— V . Z = Wk x: 2» 3 - v, V1 + k V. 5‘ #3)— Wm £9 4/(4 AWWA ,(/. > (Amok W - =0 (EUi/LEIS MW'WQ—Jgg; mﬂo M L c.» on ((\$31 (“’3’0 "LL -3_\/ . L b er X“ 3 L," W; VL’ kz' ' ‘ MWZI/‘c— ...
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