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# Introduction to Mathematical Statistics APPM 5520

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This 6 page Class Notes was uploaded by Dr. Filomena Hegmann on Thursday October 29, 2015. The Class Notes belongs to APPM 5520 at University of Colorado at Boulder taught by Jem Corcoran in Fall. Since its upload, it has received 49 views. For similar materials see /class/231868/appm-5520-university-of-colorado-at-boulder in Applied Math at University of Colorado at Boulder.

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Date Created: 10/29/15

APPM 45520 Mome Review Problems for Exam III For questions 35 38 determine whether the series is conditionally convergent absolutely convergent or divergent 35 Eilem ln lg 36 Eggnnilw 37 yaw 38 co VVT 711 ln n For questions 39 42 nd the sum of the series 22n1 39 231 5T 40 371 We 41 f1tan 1n 1 7 tan 1 n 42 220 43 Express the repeating decimal 12345345345 as a fraction 44 For What values of w does the series Zill n w converge 45 Find the sum of the series 221 7 73 correct to four decimal places 46 a Show that the series 211 is convergent b Deduce that limnnm 0 4 K1 Prove that if the series 211 an is absolutely convergent then the series Willi n1 is also absolutely convergent For questions 48 49 nd the radius of convergence and interval of convergence of the given series 7 n 2 48 220 00 n 49 2711 gins 50 Find the radius of convergence of the series 51 Find the Taylor series expansion of ne sin x at c 7r6 52 Find the Taylor series expansion of ne cos x at c 7r3 For questions 53 607 nd the Maclarin series for f and its radius of convergence Use either the direct method or the manipulation of known series Note You might nd it easiest in some cases to use the binomial series7 even though it is the subject of Section 811 53 M 54 far xliwz 55 f w ln17w 56 far wezz 58 f 59 far 1316753 gt gt gt 57 fwsinw4 gt gt 60 fw173w5 H 10 U1 9 T 90 J APPM 45520 Review Problems for Exam II Sections 7374 f dw f dy f d9 f 7 dz Solve x2 4 3L 3 ya 0 14 Zdz 112 4z f t3 dt 1t24 fxinzdw 9310 q 01 9 PO 3 H H H I HHHHHHH n True 16 6 1 and the Maclaurin series for 63 1s 63 2070 7 7 n APPM 45520 Solutions to Some Review Problems for Exam III False Example the harmonic series True by direct comparison an72 3 anti True by direct comparison False since limnH00 21 1 No info True since limH00 2 O lt 1 False Ea is smaller than 2 bn and so it may converge 3 which converges for all 3 Therefore 6 1 2200 False A power series has the form 2 3 10441170441270 a3w False It is still necessary that Rnw a 0 True 7 It should have read If Ea is divergent then 2 lanl is divergent Then the answer is True The second sum is bigger True because the coe icient of 33 is given by f 031 Or perhaps more simply you could compute f 0 by taking three derivatives of the given f and plugging in zero False Example an n and 1 in There is no question 141 Converges to 12 Converges to 5 Diverges to 00 Diverges by L Hopital s rule Diverges limit does not exist 2 O 2 H 2 I 2 03 2 2 U1 2 O 2 2 00 29 3 O 3 H 3 I r gt T Converges to zero by the squeeze theorem sin n Olt 71 and 171 a O as n a 00 Or7 don t use absolute values Both 7171 and 171 go to zero Rewrite as 1 3n 4 Converges to 634 12 Converges Rewrite as 71 lonnl and note that Ionn1 goes to zero 10 10 10 10 u10 10 10w10 10 10 10 7 S 10 i gt n1 1 2 3w10 11 12 n 11 Diverges by limit comparison test with the harmonic series Converges Limit comparison test with bn 712714 1712 Converges by alternating series test an 71 17 where bn 1n14 We have bn 3 bn and bn a O Converges by ratio test Converges by root test Diverges by n th term test for divergence sinn lt 1 lti 1n2 71n2 n2 Therefore 1 sinn Z 1n2 SZEltOO p seies with p 2 Therefore the original series is absolutely convergent and7 hence7 conver gent Converges by integral test Converges by ratio test Converges by alternating series test an inflan where bn ln Can show that bn is eventually decreasing so bn 3 bn by considering the function fw ln and its derivative Also7 need to show that bn a 0 using L Hopital s rule 3 3 gt03 Diverges by ratio test Rationalize by multiplying top and bottom by xn 1 xn 7 1 to get an 2 nxn1xnil39 Now do the limit comparison test with bn lngZ Since liniTH00 anbn 1 and Eb is a convergent p series7 the original series converges

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