Re: Curious limit

Yaroslav Bulatov wrote:
How can I find
lim_{d->infinity} x^d Gamma(d/2)/(d^(d/2))

Empirically, this limit seems to be 0 for x<=Sqrt(2e),and infinity
otherwise

I get the same answer, except for the case x = Sqrt(2e) when I get a
limit of infinity, not zero.

For brevity, denote x^d Gamma(d/2)/(d^(d/2)) by y. Using Stirling's
approximation, which should be fine in these circumstances, we can
write

y ~ x^d * sqr(2*pi) * (d/2)^(d/2 + 1/2) * exp(-d/2) / d^(d/2)
~ x^d * sqr(pi*d) * exp(-d/2) / 2^(d/2)

("~" means "approximately equal to")

Then take logs:

log(y) ~ d*log(x) + 1/2*log(pi) + 1/2*log(d) - d/2 - d/2*log(2)
~ d*k + 1/2*log(d) + 1/2*log(pi)

where k = log(x) - 1/2 - log(2)/2.

Unless k = 0, the term d*k will always dominate the term 1/2*log(d) as
d -> infinity. So,

If k > 0, that is, x > sqrt(2*e), then d*k is positive, so as d ->
infinity, log(y) -> infinity and hence y -> infinity.

If k < 0, that is, x < sqrt(2*e), then d*k is negative, so as d ->
infinity, log(y) -> -infinity and hence y -> 0.

If k = 0, that is, x = sqrt(2*e), then d*k is zero, but the term
1/2*log(d) goes to infinity anyway, so log(y) -> infinity and hence y
-> infinity.

.

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