Question

-(a) What does it mean to say that a sequence (an) is convergent, with limit L? Show that, if x and 8 are real numbers with |2 – 11 < 8 and 0<83 1/2, then 1 – 2 Hence show that, if (an) is a convergent sequence of positive real numbers with limit 1, then (1/an) is also a convergent sequence with limit 1. (b) Suppose that (bk) is a sequence of real numbers such that, for each ke N, we have 0 <bk < 3-k. For n e N, set n (n = 261 + 2?62 + ... + 2Mbn = 2%bk. k=1 Show that (cn) is a convergent sequence. (State clearly any results you use.) n2 + 2 na +3 (C) Find lim n->00 n+1 n + 2

Answer 1

convergent exists ca) say that semence (an) is L limit if for any exo there with ni EN such that lan-tce for all non can is said to be convergent semence abler certain terms an and I will be the o small with limit . if distance between arbitrarily o <8 € 1 *-!18 and Ix-less - - (x-) £ x 52

1 - - 1 - t since then *70) 11 1 < & <25 I since xet - 2 (1-7 28 proved) lim an = 1 Now let no then rom the above for S- & we have I an- 1 < & for all ages. then 11- an <a S=28 - e for allmen. lim an = 1 tanbn (b) en= 2 bit 2²b₂t then +2hbn) Coti-Co=(261+2²2 +2ht I buti) - cabi+22 bat Now Again 스 2 htl butl since osbus 3-k. Cuti-cu- anti bati 70 cut tren. for all nett Hence the sequence kennis monotonie increasing. cn=abit 22 b 2 + + 20 bn +6)2+ +639 + 3 +1+3)+ < 3 1-( (1 22 221-6" <2 for Since ang 2/1 - (3) * 1 <2 nEmten cu<2 for all nem

thence (enis bounded a bove ONOLO - secuence of real number is monotonic above then it is increasing and handed convergent therefore en is conver convergent + 2 2+2 n2-3 lim htl n+2 no n3+2n+2 n²+4-43-3n - 42-3 - lim h (n) (utz) lim n² nti (hti) rut2) 42 - nt1 lim x 42+ 3n+2 1-ħha lim It 2 + 1 2 = 1.