I follow the Spiked Math Blog and I especially liked the ‘Mathematicians Lottery‘ post which relies on the divergence of the harmonic series for the payout of the prize to insure that the payout can be managed.

The Wikipedia page on the harmonic series is very well done and the demonstation that this series diverges using an improper integral reminded me of the proof that I first saw when I was taking third semester calculs back in 1972 (forty years ago). Here is how the payout goes and the proof that you can award an arbitrary large sum of money as prizes in the lottery and get away with it even if you suppose that every ticket is a winner. Here is how all that goes:

[1] \sum_{n=1}^{\infty}\frac{1}{n}

The idea is that payout works like this: You get $1.00 the first week then (1/2)($1.00) the second week then (1/3)($1.00) the third week and because this series is divergent its partial sum is eventually greater than any finite number (even $750,000,000,000,000 “750 trillion dollars”).

The key is that the harmonic series diverges very, very slowly. To see that the series diverges at all it is enough to notice that it is bounded (from below) by something else that you can show ‘goes-to-infinity’. The choice is the following integral:

[2] {\int^{\infty}_1\frac{1}{n}\,dn}

To see why the integral bounds the series from below this picture is a great help:

bounded_from_below

Since the area under the curve from 1 to infinity is strictly less than the sum of the areas of the rectangles then if we can show that the integral diverges then that will show that the series diverges. This is because each of the rectangles has base equal to 1 and height equal to 1/n and thus area equal to 1/n and the sum of them all is exactly the value of the infinite series.

When you actually perform the integration in [2] you get:

[3] {\int^x_1\frac{1}{n}\,dn = ln(x)-ln(1)}

To see why this integrates to ln(x) you think about the inverse function theorem and the fact that the derivative of exponential is itself and the inverse of the exponential is the lograthim.

[4] {\frac{d}{dx}\left(e^x\right) = e^x}

and

[5] if {f(x) = e^x} then {f^{-1}(x) = ln(x)}

This also explains why ln(x) is an increasing function and why it is a very, very slowly increasing function (since its inverse is a very very quickly increasing function). And all this says that:

[6] \sum_{n=1}^{\infty}\frac{1}{n} > {\lim_{x\to\infty}\int^x_1\frac{1}{n}\,dn = \lim_{x\to\infty}ln(x) = \infty}

So this settles it, the harmonic series is divergent and you can always exceed an arbitrary large by just picking a greater and greater partial sum. The question then comes up just how many years do you need to wait to collect 750 trillion dollars. This comes out to be:
[7] ln(n) = \frac{(7.5)(10^{14})}{52} because there are 52 weeks in a year.

Now we just solve for n getting:

[8] n = e^{\frac{(7.5)(10^{14})}{52}}

I did visit Wolfram Alpha web site and entered the exponential and, of course, it is way beyond computation, but interestingly Wolfram Alpha displayed the following:

[9] e^{\frac{(7.5)(10^{14})}{52}} = 10^{10^{10^{1.119224798480347}}}

And this totally settles the matter as to whether or not it is OK award 750 trillion dollars to each ticket holder.

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