I was playing around and empirically identified this relation
x and n are nonnegative integers.
[; \frac{1}{(n+x+1)} = \sum_{i=0}^{n} \frac{(1)^{i}(n+x)!}{x!i!(ni)!(x+i+1)} ;]
How would I go about proving it?
How would I prove this? (some trick with factorials?)
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 jestingrabbit
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Re: How would I prove this? (some trick with factorials?)
Induction on n.
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Re: How would I prove this? (some trick with factorials?)
Hopefully not the same way you proved your username
I put the "fun" in "mathematics".
And then I took it back out.
And then I took it back out.
 gmalivuk
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Re: How would I prove this? (some trick with factorials?)
Incidentally, the [math] tags haven't been functional on here for quite some time. However, if you use the TeX The World plugin, putting ;] at the end of what you want TeXified and [; at the beginning will do the trick.
Re: How would I prove this? (some trick with factorials?)
I generally try to rewrite conglomerations of factorials as choose functions in hopes of making a counting argument, so I tried to coalesce the sea of factorials on the righthand side into choose functions. I got it as a product of these, but I never got a counting argument working, and instead I had to rely on some... intriguing choices of substitutions and identities to get the answer. A proof of the identity follows.
Spoiler:
(∫p^{2})(∫q^{2}) ≥ (∫pq)^{2}
Thanks, skeptical scientist, for knowing symbols and giving them to me.
Thanks, skeptical scientist, for knowing symbols and giving them to me.
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