Please Answer Problem #8 Problem 8 Let (an), (m) be sequences of positive number

Question

Please Answer Problem #8

Problem 8 Let (an), (m) be sequences of positive numbers, such that, for all n > 1, a) Show that, from n = 2 onwards, (m) is monotonic decreasing and (be) is monotonic increasing b) Deduce that (an) and (bn) have the same limit. Problem 9 Let (an) be a Cauchy sequence, with limit . Show that if N is such that lam- anln>N, then la - anl e for all n> N Problem 10 Let (Ln) be the sequence given by n+1 n 2 Show that (Ln) is monotonic increasing and that 1 is an upper bound, and deduce that (Ln) has a limit L, where Ls1. Problem 11 Determine the sum of the series 17 Problem 12 Investigate the convergence of the series: (n+2) n= 1 Problem 13 Use the ratio test to show that, for each fired k and each a such that 0

Explanation / Answer

a2 = (a1+b1)/2 > (a1b1) = b2

let us assume that, ak > bk  

then, ak+1 = (ak + bk)/2 > (akbk) = bk+1

hence, ak > bk implies, ak+1 > bk+1 and, a2 > b2

by the principle of mathematical induction, an > bn for all n>1

so we have, b2 < b3 < b4 < ........ < a4 < a3 < a2

therefore the sequence {an} is monotone decreasing and bounded below, and the sequence {bn} is monotone increasing and bounded above.

hence both the sequences are convergent.

let, lim an = a and lim bn = b

and, an+1 = (an+bn)/2 for all n in N

proceeding to limit as n tends to infinity , we have, a = (a+b)/2

i.e. a = b , i.e. lim an = lim bn

therefore both the sequences converge to the same limit

  

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