Problem Description Consider the following sequence of recur

Problem Description: Consider the following sequence of recurring dreams. The scenario is nearly always the same. You\'re standing at the end of a road that is 1 kilometer long, and there at the other end is an Emu, just standing there, making a funny face at you. You start to go after him, but you can only run in slow motion, about 1 meter per second. After one second the road stretches uniformly and instantaneously by 1 kilometer so now the Emu is 1998 meters away, since some of the stretch happens behind you. You try to speed up, but it feels like you are still moving in slow motion, at 1 meter per second. After another second, the roach stretches again by 1 kilometer! And this just keeps on happening. Over, and over. And then you wake up. Do you ever catch the Emu? If you do catch the Emu, how long does it take? Most of the dreams aren\'t that specific. Usually you do not know how long the road is to begin with, or how fast you are moving, or how many stretches occur. All you know is that you\'re always moving at the same slow rate, and the road stretches uniformly and instantaneously by its original amount after each second. First, make sure you understand why the Emu is 1998 meters away after the first stretch. Next, set up a sequence {dn} where dn represents the distance between you and the Emu after n seconds, but before the road does its instantaneous stretch. Then write d1= 1* (some expression involving d0) d2= 2 * (some expression involving d1) d3= 3 * (some expression involving d2) Now convert your expressions for d2 and d3 so that they only involve d0. Use this to find the general expression for dn in terms of d0. 1. Write a sentence or two explaining why the Emu is 1998 meters away from you. 2. Create expressions for d1,d2, and d3 in terms of d0 3. Use the information from #2 to write a general expression for dn in terms of d0. 4. Create a series that represents the situation that you are facing. 5. Determine if the series converges or diverges. If it converges, what does it converge to? 6. Now answer the main question. Will you catch the silly bird and if so how long will it take you?

Solution

d_0 = 1000 (in metres)

d_1 = (d_0 +1000)*(d_0 -1)/d_0 = 2000 * 999/1000 = 1998

what are we doing here ?

after n-1th second distance remaining is d_{n-1}. but the road stretches by 1000 m . if he does not move he would be d_{n-1}*1000/(n*1000) + d_{n-1} = d_{n-1}*(n+1/n) away after n seconds.

Since the total length after n-1th seconds = n*1000

Now as he moves 1 metre in nth second 1 metre expands to 1*1000/n*1000 +1 = 1*(n+1/n)

So remainig distance after n seconds = (d_{n-1}-1)*(n+1/n)

d_0 = 1000

d_1 = (1000-1) * 2 =1998

d_2 = (1998-1) * 3/2

d_n =(d_{n-1} - 1)*(n+1/n) = (d_{n-2} - 1)*(n/n-1)*(n+1/n) - n+1/n = ..... = d_0 * (n+1) - (n+1/n + n+1/n *n/n-1 + ...)

= d_0*(n+1) - (n+1/n + n+1/n-1 + n+1/n-2 ......) = d_0*(n+1) - n+1*sum_i=1 to n (1/i)

it is converging

it will meet the bird when d_n <=0

that is d_0 <= sum (1/i)