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# Test1 - (15 1(15 3(15 s.LetS=(0.1]o%— Prove MA 4633/6633...

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Unformatted text preview: (15) 1. (15) 3. (15) s. .LetS=(0.1]o{%— . Prove: MA 4633/6633 September 24‘ 2010 Test 1 Let N denote the natural numbers, N = {1,2,3... . . } and let ER denote the set of real numbers. Deﬁne a sequence (363:1 in R recursively by 51 = 1 and sn+1 = x/alsn — 2 for every n. 2 1. Show by induction that for every n E N, 3n S 5n+1 S 4- %:n E N}. Find infS and sup 3. (No proof is required.) For each n E N, deﬁne en 2 sin2 (7*); i.e., (5,1)”? m ( deﬁnition of limit to show that He»: nb-Iuﬂ P vii-Ito LIN , ,0, . . . ). Use the negation of the \$ lim 3.” 7E g. cos(n) 2n+3—>Uasn-—>oo. Prove: If (;r.n)2_°=1 C R is bounded and nondecreasing, then lim can = sup 3:”. n—roo 7121 Problems (1) and (5) imply that the sequence 31 = 1 and sn+1 = V’zlsn — 2 (for all n 2 1) is convergent. Use properties of limits to ﬁnd iim 3n. (Notice that. en 2 1 for all n.) n—poo Is. 50% \$9.4 / m CFC—LA“; (K‘s/{m \711) ,1 =57 ENeN SJ. h+t Lgﬁc-leﬂc‘. QLiCJL/ﬂ. ’h/UAAI 1‘3 ~45: Jr‘r-Sq J ’hx‘E/k S 4 m1 '5. (“I g. l: (1.) S Sphinx-*1 I5. lngrcggtq> CL L] A I: 1... ”79¢. m& Saga-L. Q—5 r\ Sn+1;i"fsn-—-7— {WM-5 l \l/ N— V a- :W: ...
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