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Consider the sequence akdefined recursively bya1=1and fork>1,ak=2ak-1.Prove thatakโ†’2by first proving that the limit must exist. (Hint: Use induction to show that the terms of the sequence may be expressed with the closed formula forak=21-12k-1)

Short Answer

Expert verified

The value oflimkโ†’โˆžak=2

Step by step solution

01

Step 1. Given information

Consider the given sequence akdefined recursively bya1=1a nd fork>1,ak=2ak-1

02

Step 2. Finding the values of  terms a1,a2,a3

The terms of the sequence is defined as

a1=1.............(1)a2=2a1(putk=2inak=2ak-1)a2=2(simplify).................(2)

From equations,(1) and (2) ,it is observed that

0<a1<a2<2(because2<2)

Put k=3in ak=2ak-1to get

=22(substitutea2=2)

Thusa2<a3<2(because22<2)

03

Step 3. Finding the given sequence is bounded or not.

The general term of the sequence is defined as

ak=222.......(k-1)times

countinuing likewisw,the following inequality is obtained

0<a1<a2<a3<.......<ak<2

The sequenceak is an increasing sequence because

0<a1<a2<a3<.......<ak

the sequence akhas a lower bound and upper bound.Thus,the given sequence is bounded

04

Step 4. Find the value of limkโ†’โˆžak

The monotonic increasing sequence which is bounded above is convergent.The given sequence akdefined recursively by a1=1and for k>1,ak=2ak-1is increasing is bounded above by 2. thus,the sequence is convergent.

Assume the limit of the sequence akisl.

Therefore,

limkโ†’โˆžak=limkโ†’โˆž2ak-1l=2limkโ†’โˆžak-2(becauseakโ†’1)l=2ll2-2l=0(squaring)l(l-2)=0(factorize)l=0,2(solveforl)

The sequence cannot converge to 0 because a1=1 and the sequence is increasing.Therefore,the value of lis 2

Thus, the value of limkโ†’โˆžak=2

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Most popular questions from this chapter

Consider the series

Fill in the blanks and select the correct word:

Iflimkโ†’โˆžakbk=โˆžandโˆ‘k=1โˆž_____divergesthenโˆ‘k=1โˆž_____(converges/diverges).

Prove Theorem 7.24 (a). That is, show that if c is a real number andโˆ‘k=1โˆžak is a convergent series, then โˆ‘k=1โˆžcak=cโˆ‘k=1โˆžak.

Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If akโ†’0, then โˆ‘k=1โˆžakconverges.

(b) True or False: If โˆ‘k=1โˆžakconverges, then akโ†’0.

(c) True or False: The improper integral โˆซ1โˆžf(x)dxconverges if and only if the series โˆ‘k=1โˆžf(k)converges.

(d) True or False: The harmonic series converges.

(e) True or False: If p>1, the series โˆ‘k=1โˆžk-pconverges.

(f) True or False: If f(x)โ†’0as xโ†’โˆž, then โˆ‘k=1โˆžf(k) converges.

(g) True or False: If โˆ‘k=1โˆžf(k)converges, then f(x)โ†’0as xโ†’โˆž.

(h) True or False: If โˆ‘k=1โˆžak=Land {Sn}is the sequence of partial sums for the series, then the sequence of remainders {L-Sn}converges to 0.

Let โˆ‘k=1โˆžakbeaconvergentseriesandโˆ‘k=1โˆžbkbeadivergentseries.Prove that the series โˆ‘k=1โˆžak+bkdiverges.

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