- #1
Vespero
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- 0
Homework Statement
Define the numbers
[tex] G_n = \prod_{k=1}^n (\prod_{j=1}^{k-1}\frac{k}{j}).[/tex]
(a) Show that [itex]G_n[/itex] is an integer, [itex]n>1[/itex];
(b) Show that for each prime [itex]p[/itex], there are infinitely many [itex]n>1[/itex] such that [itex]p[/itex] does not divide [itex]G_n.[/itex]
Homework Equations
The Attempt at a Solution
I can see that the expansion is
[tex] G_n = (\prod_{j=1}^{0}\frac{k}{j}) (\dfrac{2}{1})(\dfrac{3}{1}\dfrac{3}{2})...(\dfrac{n}{1}\dfrac{n}{2}...\dfrac{n}{n-1}).[/tex]
However, what happens when n = 1, or more specifically, in the first factor of the general expansion where k = 1? Do we simply ignore [itex]\prod_{j=1}^{0}\frac{k}{j}[/itex]? Does it equal 1? I assume it doesn't equal 0, or the function would always equal 0. Once I understand this, I might actually be able to work the problem.