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Suppose a light bulb is turned on at time $t=0.$ It switches off at $t=1,$ on again at $t=1+{1 \over 2},$ off at $t=1+ {1 \over 2}+{1 \over 3},$ and so forth. As $t$ goes to $\infty,$ what proportion of the time is the light bulb on?

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3 Answers 3

up vote 1 down vote accepted

The whole series is the alternating harmonic series, which in the limit as n approaches infinity equals $ln 2$.

As both individual series diverge, the difference goes to a 0 fraction in the limit, and the ratio approaches 1.0. Therefore the light is on half the time.

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Consider the first few terms of the series concerning the period of time the light is turned on and off:

$$\text{ON: }1, \frac 13, \frac 15,...\\ \text{OFF: } \frac 12,\frac 14, \frac 16,...$$

You can clearly see that the times you are looking for are:

$$\displaystyle\sum^{\infty}_{i=1} {\frac 1{2i-1}} $$

and

$$\displaystyle\sum^{\infty}_{i=1} {\frac 1{2i}} $$

Use harmonic series to solve

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So....what proportion of the time is the light bulb on? Certainly this diverges. –  Don Larynx Oct 9 '13 at 2:49
    
The interesting thing is that the difference of the two series does converge, as the accepted answer points out. –  soakley Oct 9 '13 at 22:56
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At any point in time, the light has been on longer than it has been off, as $1 > \frac{1}{2}$, $\frac{1}{3} > \frac{1}{4}$, etc. If we start looking at the light at $t = 1$, then at any point in time the light has been off longer than it has been on, as $\frac{1}{2} > \frac{1}{3}$, $\frac{1}{4} > \frac{1}{5}$, etc. So the proportion of the time that the light is on must be $\frac{1}{2}$.

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That's spectacularly unconvincing. –  Pieter Geerkens Oct 9 '13 at 3:27
    
How so? If we denote the proportion of the time that the light is on by $P$, then the first sentence implies $P \geq \frac{1}{2}$ and the second sentence implies $P \leq \frac{1}{2}$. –  Arthur Oct 9 '13 at 4:14
    
It makes sense to me, but only after I understood Pieter's answer, which isn't that clear on first reading, either. But essentially Arthur is arguing the 1 at the beginning simply doesn't matter, and he is correct. –  soakley Oct 9 '13 at 22:55
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