Editing Harmonic series (mathematics) (section)

===Crossing a desert===
{{main|Jeep problem}}
[[File:Jeep problem 1.png|thumb|Solution to the jeep problem for <math>n=3</math>, showing the amount of fuel in each depot and in the jeep at each step]]
The [[jeep problem]] or desert-crossing problem is included in a 9th-century problem collection by [[Alcuin]], ''[[Propositiones ad Acuendos Juvenes]]'' (formulated in terms of camels rather than jeeps), but with an incorrect solution.{{r|alcuin}} The problem asks how far into the desert a jeep can travel and return, starting from a base with <math>n</math> loads of fuel, by carrying some of the fuel into the desert and leaving it in depots. The optimal solution involves placing depots spaced at distances <math>\tfrac{r}{2n}, \tfrac{r}{2(n-1)}, \tfrac{r}{2(n-2)}, \dots</math> from the starting point and each other, where <math>r</math> is the range of distance that the jeep can travel with a single load of fuel. On each trip out and back from the base, the jeep places one more depot, refueling at the other depots along the way, and placing as much fuel as it can in the newly placed depot while still leaving enough for itself to return to the previous depots and the base. Therefore, the total distance reached on the <math>n</math>th trip is <math display=block>\frac{r}{2n}+\frac{r}{2(n-1)}+\frac{r}{2(n-2)}+\cdots=\frac{r}{2} H_n,</math>
where <math>H_n</math> is the {{nowrap|<math>n</math>th}} harmonic number. The divergence of the harmonic series implies that crossings of any length are possible with enough fuel.{{r|gale}}

For instance, for Alcuin's version of the problem, <math>r=30</math>: a camel can carry 30 measures of grain and can travel one leuca while eating a single measure, where a leuca is a unit of distance roughly equal to {{convert|2.3|km|mi}}. The problem has <math>n=3</math>: there are 90 measures of grain, enough to supply three trips. For the standard formulation of the desert-crossing problem, it would be possible for the camel to travel <math>\tfrac{30}{2}\bigl(\tfrac13+\tfrac12+\tfrac11)=27.5</math> leucas and return, by placing a grain storage depot 5 leucas from the base on the first trip and 12.5 leucas from the base on the second trip. However, Alcuin instead asks a slightly different question, how much grain can be transported a distance of 30 leucas without a final return trip, and either strands some camels in the desert or fails to account for the amount of grain consumed by a camel on its return trips.{{r|alcuin}}