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Question

None of the following experiments can be carried out in practice.

But that doesn't stop us from thinking about them.

In one way or another each of them involves the infinite series

1/2 + 1/4 + 1/8 + ... + = 1.

Each asks for a response. Which responses, if any, can be given?

  1. A lamp is switched on and remains on for 1/2 minute.
    Then it's switched off for 1/4 minute.
    Then on for 1/8 minute, and so on.
    After 1 minute, of course, the switching stops.
    Describe the condition [on or off] of the lamp.
    .
  2. A supercomputer is programmed to calculate a print the value of pi, one digit at at time.
    The first digit prints after 1/2 minute. Each succeeding digit prints in 1/2 the time, and,
    [to keep the supply of printer paper finite], in 1/2 the space of the previous digit.
    After 1 minute, of course, the printing stops.
    Can we at last determine pi's final digit?
    .
  3. A marble rests in tray A.
    After 1/4 minute a machine transfers the marble to tray B and rests for another
    1/4 minute while a different machine transfers the marble back to tray A.
    During the next 1/8 minute the first machine puts the marble back to tray B
    and rests 1/8 minute while the second machine returns it to tray A.
    After 1 minute the halved-time transferring process stops.
    Describe the location [tray A or B] of the marble.
    .
  4. A constant-speed runner completes 1/2 of his journey in 1/4 minute; then he rests for 1/4 minute.
    In the next 1/8 minute he completes 1/4 of his journey, resting thereafter for 1/8 minute. And so on.
    Describe the position of the runner after 1 minute.
    .
  5. A certain toy ball returns to 1/2 its original height 1/2 minute after being dropped.
    It falls from that point and returns to 1/4 the original height 1/4 minute later,
    and to 1/8 the height 1/8 minute after that. And so on.
    Describe the motion of the ball 1 minute after the initial drop.
    .
  6. The ball in question 5 is initially red.
    It changes color, alternating between red and blue, each time it strikes the ground.
    Describe its color after 1 minute.
    .

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i havent read all the responses but theoretically

numero dos'o

if the computer could reach an infinite speed yes no matter if there is one or not

either the computer will not have the speed to finish. or it will print it out.

for example say pi has 200 digits the computer would need to be able to print and think of a digit in (1/2)^200s

if it cant do this it would never reach that digit even though this pi is supposedly finite

as the digits increase by one the computers needed speed doubles. so even if pi has infinite digits. if the computers speed is 2^infinite it wins. This is how the incredibly annoying (yet useful) concept of infinite works

so the question is really if a computer's speed could be infinite would the world explode

I vote yes on the explosion. 'cept it would be the universe.

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