Closed form of the limit of a sequence (weighted average)

Closed form of the limit of a sequence (weighted average)

I have a sequence, which can actually be seen as Riemann-Stieltjes
integration with a binomial distribution. $\rho \in (0,1)$.
$$ S_N
:=‡"_{n=0}^{N}ƒÏ^{N-n}(1-ƒÏ)^{n}\binom{N}{n}\left(\frac{n}{N}\right)^\theta
$$
Using convexity/concavity of $(n/N)^\theta$ I can show the sequence
declines/increases with N. The sequence is also bounded between 0 and 1. I
wonder if there is a closed form limit of this sequence.
I appreciate any thoughts on this.